Hilbert's nineteenth problem (English Wikipedia)

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See (Hilbert 1900) or, equivalently, one of its translations. Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
"Sind die Lösungen regulärer Variationsprobleme stets notwendig analytisch?" (English translation by Mary Frances Winston Newson:-"Are the solutions of regular problems in the calculus of variations always necessarily analytic?"), formulating the problem with the same words of Hilbert (1900, p. 288). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
See (Hilbert 1900, pp. 288–289), or the corresponding section on the nineteenth problem in any of its translations or reprints, or the subsection "The origins of the problem" in the historical section of this entry. Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
Hilbert does not cite explicitly Joseph Liouville and considers the constant Gaussian curvature $K$ as equal to $-1/2$ : compare the relevant entry with (Hilbert 1900, p. 288). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
Unlike Liouville's work, Picard's work is explicitly cited by Hilbert (1900, p. 288 and footnote 1 in the same page). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
See (Hilbert 1900, p. 288). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
In his exact words: "Reguläres Variationsproblem". Hilbert's definition of a regular variational problem is stronger than the one currently used, for example, in (Gilbarg & Trudinger 2001, p. 289). Gilbarg, David; Trudinger, Neil S. (2001) [1998], Elliptic partial differential equations of second order, Classics in Mathematics (Revised 3rd printing of 2nd ed.), Berlin – Heidelberg – New York: Springer Verlag, pp. xiv+517, ISBN 978-3-540-41160-4, MR 1814364, Zbl 1042.35002.
English translation by Mary Frances Winston Newson: Hilbert's (1900, p. 288) precise words are:-"... d. h. ob jede Lagrangesche partielle Differentialgleichung eines reguläres Variationsproblem die Eigenschaft at, daß sie nur analytische Integrale zuläßt" (Italics emphasis by Hilbert himself). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
See (Giaquinta 1983, p. 59), (Giusti 1994, p. 7 footnote 7 and p. 353), (Gohberg 1999, p. 1), (Hedberg 1999, pp. 10–11), (Kristensen & Mingione 2011, p. 5 and p. 8), and (Mingione 2006, p. 368). Giaquinta, Mariano (1983), Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton, New Jersey: Princeton University Press, pp. vii+297, ISBN 978-0-691-08330-8, MR 0717034, Zbl 0516.49003. Giusti, Enrico (1994), Metodi diretti nel calcolo delle variazioni, Monografie Matematiche (in Italian), Bologna: Unione Matematica Italiana, pp. VI+422, MR 1707291, Zbl 0942.49002, translated in English as Giusti, Enrico (2003), Direct Methods in the Calculus of Variations, River Edge, New Jersey – London – Singapore: World Scientific Publishing, pp. viii+403, doi:10.1142/9789812795557, ISBN 978-981-238-043-2, MR 1962933, Zbl 1028.49001. Gohberg, Israel (1999), "Vladimir Maz'ya: Friend and Mathematician. Recollections", in Rossman, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya anniversary collection. Vol. 1: On Maz'ya's work in functional analysis, partial differential equations and applications. Based on talks given at the conference, Rostock, Germany, August 31 – September 4, 1998, Operator Theory. Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 1–5, ISBN 978-3-7643-6201-0, MR 1747861, Zbl 0939.01018. Hedberg, Lars Inge (1999), "On Maz'ya's work in potential theory and the theory of function spaces", in Rossmann, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya Anniversary Collection, Operator Theory: Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 7–16, doi:10.1007/978-3-0348-8675-8_2, ISBN 978-3-0348-9726-6, MR 1747862, Zbl 0939.31001 Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.
See (Giaquinta 1983, pp. 54–59), (Giusti 1994, p. 7 and pp. 353). Giaquinta, Mariano (1983), Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton, New Jersey: Princeton University Press, pp. vii+297, ISBN 978-0-691-08330-8, MR 0717034, Zbl 0516.49003. Giusti, Enrico (1994), Metodi diretti nel calcolo delle variazioni, Monografie Matematiche (in Italian), Bologna: Unione Matematica Italiana, pp. VI+422, MR 1707291, Zbl 0942.49002, translated in English as Giusti, Enrico (2003), Direct Methods in the Calculus of Variations, River Edge, New Jersey – London – Singapore: World Scientific Publishing, pp. viii+403, doi:10.1142/9789812795557, ISBN 978-981-238-043-2, MR 1962933, Zbl 1028.49001.
See (Hedberg 1999, pp. 10–11), (Kristensen & Mingione 2011, p. 5 and p. 8) and (Mingione 2006, p. 368). Hedberg, Lars Inge (1999), "On Maz'ya's work in potential theory and the theory of function spaces", in Rossmann, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya Anniversary Collection, Operator Theory: Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 7–16, doi:10.1007/978-3-0348-8675-8_2, ISBN 978-3-0348-9726-6, MR 1747862, Zbl 0939.31001 Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.
According to (Gohberg 1999, p. 1). Gohberg, Israel (1999), "Vladimir Maz'ya: Friend and Mathematician. Recollections", in Rossman, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya anniversary collection. Vol. 1: On Maz'ya's work in functional analysis, partial differential equations and applications. Based on talks given at the conference, Rostock, Germany, August 31 – September 4, 1998, Operator Theory. Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 1–5, ISBN 978-3-7643-6201-0, MR 1747861, Zbl 0939.01018.
See (Giaquinta 1983, pp. 54–59) and (Giusti 1994, p. 7, pp. 202–203 and pp. 317–318). Giaquinta, Mariano (1983), Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton, New Jersey: Princeton University Press, pp. vii+297, ISBN 978-0-691-08330-8, MR 0717034, Zbl 0516.49003. Giusti, Enrico (1994), Metodi diretti nel calcolo delle variazioni, Monografie Matematiche (in Italian), Bologna: Unione Matematica Italiana, pp. VI+422, MR 1707291, Zbl 0942.49002, translated in English as Giusti, Enrico (2003), Direct Methods in the Calculus of Variations, River Edge, New Jersey – London – Singapore: World Scientific Publishing, pp. viii+403, doi:10.1142/9789812795557, ISBN 978-981-238-043-2, MR 1962933, Zbl 1028.49001.
For more information about the work of Jindřich Nečas see the work of Kristensen & Mingione (2011, §3.3, pp. 9–12) and (Mingione 2006, §3.3, pp. 369–370). Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.

archive.org

See (Hilbert 1900) or, equivalently, one of its translations. Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
"Sind die Lösungen regulärer Variationsprobleme stets notwendig analytisch?" (English translation by Mary Frances Winston Newson:-"Are the solutions of regular problems in the calculus of variations always necessarily analytic?"), formulating the problem with the same words of Hilbert (1900, p. 288). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
See (Hilbert 1900, pp. 288–289), or the corresponding section on the nineteenth problem in any of its translations or reprints, or the subsection "The origins of the problem" in the historical section of this entry. Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
Hilbert does not cite explicitly Joseph Liouville and considers the constant Gaussian curvature $K$ as equal to $-1/2$ : compare the relevant entry with (Hilbert 1900, p. 288). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
Unlike Liouville's work, Picard's work is explicitly cited by Hilbert (1900, p. 288 and footnote 1 in the same page). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
See (Hilbert 1900, p. 288). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
English translation by Mary Frances Winston Newson: Hilbert's (1900, p. 288) precise words are:-"... d. h. ob jede Lagrangesche partielle Differentialgleichung eines reguläres Variationsproblem die Eigenschaft at, daß sie nur analytische Integrale zuläßt" (Italics emphasis by Hilbert himself). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.

books.google.com

In his exact words: "Reguläres Variationsproblem". Hilbert's definition of a regular variational problem is stronger than the one currently used, for example, in (Gilbarg & Trudinger 2001, p. 289). Gilbarg, David; Trudinger, Neil S. (2001) [1998], Elliptic partial differential equations of second order, Classics in Mathematics (Revised 3rd printing of 2nd ed.), Berlin – Heidelberg – New York: Springer Verlag, pp. xiv+517, ISBN 978-3-540-41160-4, MR 1814364, Zbl 1042.35002.
See (Giaquinta 1983, p. 59), (Giusti 1994, p. 7 footnote 7 and p. 353), (Gohberg 1999, p. 1), (Hedberg 1999, pp. 10–11), (Kristensen & Mingione 2011, p. 5 and p. 8), and (Mingione 2006, p. 368). Giaquinta, Mariano (1983), Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton, New Jersey: Princeton University Press, pp. vii+297, ISBN 978-0-691-08330-8, MR 0717034, Zbl 0516.49003. Giusti, Enrico (1994), Metodi diretti nel calcolo delle variazioni, Monografie Matematiche (in Italian), Bologna: Unione Matematica Italiana, pp. VI+422, MR 1707291, Zbl 0942.49002, translated in English as Giusti, Enrico (2003), Direct Methods in the Calculus of Variations, River Edge, New Jersey – London – Singapore: World Scientific Publishing, pp. viii+403, doi:10.1142/9789812795557, ISBN 978-981-238-043-2, MR 1962933, Zbl 1028.49001. Gohberg, Israel (1999), "Vladimir Maz'ya: Friend and Mathematician. Recollections", in Rossman, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya anniversary collection. Vol. 1: On Maz'ya's work in functional analysis, partial differential equations and applications. Based on talks given at the conference, Rostock, Germany, August 31 – September 4, 1998, Operator Theory. Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 1–5, ISBN 978-3-7643-6201-0, MR 1747861, Zbl 0939.01018. Hedberg, Lars Inge (1999), "On Maz'ya's work in potential theory and the theory of function spaces", in Rossmann, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya Anniversary Collection, Operator Theory: Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 7–16, doi:10.1007/978-3-0348-8675-8_2, ISBN 978-3-0348-9726-6, MR 1747862, Zbl 0939.31001 Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.
See (Giaquinta 1983, pp. 54–59), (Giusti 1994, p. 7 and pp. 353). Giaquinta, Mariano (1983), Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton, New Jersey: Princeton University Press, pp. vii+297, ISBN 978-0-691-08330-8, MR 0717034, Zbl 0516.49003. Giusti, Enrico (1994), Metodi diretti nel calcolo delle variazioni, Monografie Matematiche (in Italian), Bologna: Unione Matematica Italiana, pp. VI+422, MR 1707291, Zbl 0942.49002, translated in English as Giusti, Enrico (2003), Direct Methods in the Calculus of Variations, River Edge, New Jersey – London – Singapore: World Scientific Publishing, pp. viii+403, doi:10.1142/9789812795557, ISBN 978-981-238-043-2, MR 1962933, Zbl 1028.49001.
According to (Gohberg 1999, p. 1). Gohberg, Israel (1999), "Vladimir Maz'ya: Friend and Mathematician. Recollections", in Rossman, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya anniversary collection. Vol. 1: On Maz'ya's work in functional analysis, partial differential equations and applications. Based on talks given at the conference, Rostock, Germany, August 31 – September 4, 1998, Operator Theory. Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 1–5, ISBN 978-3-7643-6201-0, MR 1747861, Zbl 0939.01018.
See (Giaquinta 1983, pp. 54–59) and (Giusti 1994, p. 7, pp. 202–203 and pp. 317–318). Giaquinta, Mariano (1983), Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton, New Jersey: Princeton University Press, pp. vii+297, ISBN 978-0-691-08330-8, MR 0717034, Zbl 0516.49003. Giusti, Enrico (1994), Metodi diretti nel calcolo delle variazioni, Monografie Matematiche (in Italian), Bologna: Unione Matematica Italiana, pp. VI+422, MR 1707291, Zbl 0942.49002, translated in English as Giusti, Enrico (2003), Direct Methods in the Calculus of Variations, River Edge, New Jersey – London – Singapore: World Scientific Publishing, pp. viii+403, doi:10.1142/9789812795557, ISBN 978-981-238-043-2, MR 1962933, Zbl 1028.49001.

dml.cz

See (Giaquinta 1983, p. 59), (Giusti 1994, p. 7 footnote 7 and p. 353), (Gohberg 1999, p. 1), (Hedberg 1999, pp. 10–11), (Kristensen & Mingione 2011, p. 5 and p. 8), and (Mingione 2006, p. 368). Giaquinta, Mariano (1983), Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton, New Jersey: Princeton University Press, pp. vii+297, ISBN 978-0-691-08330-8, MR 0717034, Zbl 0516.49003. Giusti, Enrico (1994), Metodi diretti nel calcolo delle variazioni, Monografie Matematiche (in Italian), Bologna: Unione Matematica Italiana, pp. VI+422, MR 1707291, Zbl 0942.49002, translated in English as Giusti, Enrico (2003), Direct Methods in the Calculus of Variations, River Edge, New Jersey – London – Singapore: World Scientific Publishing, pp. viii+403, doi:10.1142/9789812795557, ISBN 978-981-238-043-2, MR 1962933, Zbl 1028.49001. Gohberg, Israel (1999), "Vladimir Maz'ya: Friend and Mathematician. Recollections", in Rossman, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya anniversary collection. Vol. 1: On Maz'ya's work in functional analysis, partial differential equations and applications. Based on talks given at the conference, Rostock, Germany, August 31 – September 4, 1998, Operator Theory. Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 1–5, ISBN 978-3-7643-6201-0, MR 1747861, Zbl 0939.01018. Hedberg, Lars Inge (1999), "On Maz'ya's work in potential theory and the theory of function spaces", in Rossmann, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya Anniversary Collection, Operator Theory: Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 7–16, doi:10.1007/978-3-0348-8675-8_2, ISBN 978-3-0348-9726-6, MR 1747862, Zbl 0939.31001 Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.
See (Hedberg 1999, pp. 10–11), (Kristensen & Mingione 2011, p. 5 and p. 8) and (Mingione 2006, p. 368). Hedberg, Lars Inge (1999), "On Maz'ya's work in potential theory and the theory of function spaces", in Rossmann, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya Anniversary Collection, Operator Theory: Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 7–16, doi:10.1007/978-3-0348-8675-8_2, ISBN 978-3-0348-9726-6, MR 1747862, Zbl 0939.31001 Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.
For more information about the work of Jindřich Nečas see the work of Kristensen & Mingione (2011, §3.3, pp. 9–12) and (Mingione 2006, §3.3, pp. 369–370). Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.

doi.org

See (Hilbert 1900) or, equivalently, one of its translations. Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
"Sind die Lösungen regulärer Variationsprobleme stets notwendig analytisch?" (English translation by Mary Frances Winston Newson:-"Are the solutions of regular problems in the calculus of variations always necessarily analytic?"), formulating the problem with the same words of Hilbert (1900, p. 288). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
See (Hilbert 1900, pp. 288–289), or the corresponding section on the nineteenth problem in any of its translations or reprints, or the subsection "The origins of the problem" in the historical section of this entry. Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
Hilbert does not cite explicitly Joseph Liouville and considers the constant Gaussian curvature $K$ as equal to $-1/2$ : compare the relevant entry with (Hilbert 1900, p. 288). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
Unlike Liouville's work, Picard's work is explicitly cited by Hilbert (1900, p. 288 and footnote 1 in the same page). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
See (Hilbert 1900, p. 288). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
English translation by Mary Frances Winston Newson: Hilbert's (1900, p. 288) precise words are:-"... d. h. ob jede Lagrangesche partielle Differentialgleichung eines reguläres Variationsproblem die Eigenschaft at, daß sie nur analytische Integrale zuläßt" (Italics emphasis by Hilbert himself). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
See (Giaquinta 1983, p. 59), (Giusti 1994, p. 7 footnote 7 and p. 353), (Gohberg 1999, p. 1), (Hedberg 1999, pp. 10–11), (Kristensen & Mingione 2011, p. 5 and p. 8), and (Mingione 2006, p. 368). Giaquinta, Mariano (1983), Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton, New Jersey: Princeton University Press, pp. vii+297, ISBN 978-0-691-08330-8, MR 0717034, Zbl 0516.49003. Giusti, Enrico (1994), Metodi diretti nel calcolo delle variazioni, Monografie Matematiche (in Italian), Bologna: Unione Matematica Italiana, pp. VI+422, MR 1707291, Zbl 0942.49002, translated in English as Giusti, Enrico (2003), Direct Methods in the Calculus of Variations, River Edge, New Jersey – London – Singapore: World Scientific Publishing, pp. viii+403, doi:10.1142/9789812795557, ISBN 978-981-238-043-2, MR 1962933, Zbl 1028.49001. Gohberg, Israel (1999), "Vladimir Maz'ya: Friend and Mathematician. Recollections", in Rossman, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya anniversary collection. Vol. 1: On Maz'ya's work in functional analysis, partial differential equations and applications. Based on talks given at the conference, Rostock, Germany, August 31 – September 4, 1998, Operator Theory. Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 1–5, ISBN 978-3-7643-6201-0, MR 1747861, Zbl 0939.01018. Hedberg, Lars Inge (1999), "On Maz'ya's work in potential theory and the theory of function spaces", in Rossmann, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya Anniversary Collection, Operator Theory: Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 7–16, doi:10.1007/978-3-0348-8675-8_2, ISBN 978-3-0348-9726-6, MR 1747862, Zbl 0939.31001 Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.
See (Giaquinta 1983, pp. 54–59), (Giusti 1994, p. 7 and pp. 353). Giaquinta, Mariano (1983), Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton, New Jersey: Princeton University Press, pp. vii+297, ISBN 978-0-691-08330-8, MR 0717034, Zbl 0516.49003. Giusti, Enrico (1994), Metodi diretti nel calcolo delle variazioni, Monografie Matematiche (in Italian), Bologna: Unione Matematica Italiana, pp. VI+422, MR 1707291, Zbl 0942.49002, translated in English as Giusti, Enrico (2003), Direct Methods in the Calculus of Variations, River Edge, New Jersey – London – Singapore: World Scientific Publishing, pp. viii+403, doi:10.1142/9789812795557, ISBN 978-981-238-043-2, MR 1962933, Zbl 1028.49001.
See (Hedberg 1999, pp. 10–11), (Kristensen & Mingione 2011, p. 5 and p. 8) and (Mingione 2006, p. 368). Hedberg, Lars Inge (1999), "On Maz'ya's work in potential theory and the theory of function spaces", in Rossmann, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya Anniversary Collection, Operator Theory: Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 7–16, doi:10.1007/978-3-0348-8675-8_2, ISBN 978-3-0348-9726-6, MR 1747862, Zbl 0939.31001 Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.
See (Giaquinta 1983, pp. 54–59) and (Giusti 1994, p. 7, pp. 202–203 and pp. 317–318). Giaquinta, Mariano (1983), Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton, New Jersey: Princeton University Press, pp. vii+297, ISBN 978-0-691-08330-8, MR 0717034, Zbl 0516.49003. Giusti, Enrico (1994), Metodi diretti nel calcolo delle variazioni, Monografie Matematiche (in Italian), Bologna: Unione Matematica Italiana, pp. VI+422, MR 1707291, Zbl 0942.49002, translated in English as Giusti, Enrico (2003), Direct Methods in the Calculus of Variations, River Edge, New Jersey – London – Singapore: World Scientific Publishing, pp. viii+403, doi:10.1142/9789812795557, ISBN 978-981-238-043-2, MR 1962933, Zbl 1028.49001.
For more information about the work of Jindřich Nečas see the work of Kristensen & Mingione (2011, §3.3, pp. 9–12) and (Mingione 2006, §3.3, pp. 369–370). Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.

handle.net

hdl.handle.net

See (Giaquinta 1983, p. 59), (Giusti 1994, p. 7 footnote 7 and p. 353), (Gohberg 1999, p. 1), (Hedberg 1999, pp. 10–11), (Kristensen & Mingione 2011, p. 5 and p. 8), and (Mingione 2006, p. 368). Giaquinta, Mariano (1983), Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton, New Jersey: Princeton University Press, pp. vii+297, ISBN 978-0-691-08330-8, MR 0717034, Zbl 0516.49003. Giusti, Enrico (1994), Metodi diretti nel calcolo delle variazioni, Monografie Matematiche (in Italian), Bologna: Unione Matematica Italiana, pp. VI+422, MR 1707291, Zbl 0942.49002, translated in English as Giusti, Enrico (2003), Direct Methods in the Calculus of Variations, River Edge, New Jersey – London – Singapore: World Scientific Publishing, pp. viii+403, doi:10.1142/9789812795557, ISBN 978-981-238-043-2, MR 1962933, Zbl 1028.49001. Gohberg, Israel (1999), "Vladimir Maz'ya: Friend and Mathematician. Recollections", in Rossman, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya anniversary collection. Vol. 1: On Maz'ya's work in functional analysis, partial differential equations and applications. Based on talks given at the conference, Rostock, Germany, August 31 – September 4, 1998, Operator Theory. Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 1–5, ISBN 978-3-7643-6201-0, MR 1747861, Zbl 0939.01018. Hedberg, Lars Inge (1999), "On Maz'ya's work in potential theory and the theory of function spaces", in Rossmann, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya Anniversary Collection, Operator Theory: Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 7–16, doi:10.1007/978-3-0348-8675-8_2, ISBN 978-3-0348-9726-6, MR 1747862, Zbl 0939.31001 Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.
See (Hedberg 1999, pp. 10–11), (Kristensen & Mingione 2011, p. 5 and p. 8) and (Mingione 2006, p. 368). Hedberg, Lars Inge (1999), "On Maz'ya's work in potential theory and the theory of function spaces", in Rossmann, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya Anniversary Collection, Operator Theory: Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 7–16, doi:10.1007/978-3-0348-8675-8_2, ISBN 978-3-0348-9726-6, MR 1747862, Zbl 0939.31001 Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.
For more information about the work of Jindřich Nečas see the work of Kristensen & Mingione (2011, §3.3, pp. 9–12) and (Mingione 2006, §3.3, pp. 369–370). Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.

mathunion.org

See (Hilbert 1900) or, equivalently, one of its translations. Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
"Sind die Lösungen regulärer Variationsprobleme stets notwendig analytisch?" (English translation by Mary Frances Winston Newson:-"Are the solutions of regular problems in the calculus of variations always necessarily analytic?"), formulating the problem with the same words of Hilbert (1900, p. 288). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
See (Hilbert 1900, pp. 288–289), or the corresponding section on the nineteenth problem in any of its translations or reprints, or the subsection "The origins of the problem" in the historical section of this entry. Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
Hilbert does not cite explicitly Joseph Liouville and considers the constant Gaussian curvature $K$ as equal to $-1/2$ : compare the relevant entry with (Hilbert 1900, p. 288). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
Unlike Liouville's work, Picard's work is explicitly cited by Hilbert (1900, p. 288 and footnote 1 in the same page). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
See (Hilbert 1900, p. 288). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
English translation by Mary Frances Winston Newson: Hilbert's (1900, p. 288) precise words are:-"... d. h. ob jede Lagrangesche partielle Differentialgleichung eines reguläres Variationsproblem die Eigenschaft at, daß sie nur analytische Integrale zuläßt" (Italics emphasis by Hilbert himself). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.

ox.ac.uk

maths.ox.ac.uk

See (Giaquinta 1983, p. 59), (Giusti 1994, p. 7 footnote 7 and p. 353), (Gohberg 1999, p. 1), (Hedberg 1999, pp. 10–11), (Kristensen & Mingione 2011, p. 5 and p. 8), and (Mingione 2006, p. 368). Giaquinta, Mariano (1983), Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton, New Jersey: Princeton University Press, pp. vii+297, ISBN 978-0-691-08330-8, MR 0717034, Zbl 0516.49003. Giusti, Enrico (1994), Metodi diretti nel calcolo delle variazioni, Monografie Matematiche (in Italian), Bologna: Unione Matematica Italiana, pp. VI+422, MR 1707291, Zbl 0942.49002, translated in English as Giusti, Enrico (2003), Direct Methods in the Calculus of Variations, River Edge, New Jersey – London – Singapore: World Scientific Publishing, pp. viii+403, doi:10.1142/9789812795557, ISBN 978-981-238-043-2, MR 1962933, Zbl 1028.49001. Gohberg, Israel (1999), "Vladimir Maz'ya: Friend and Mathematician. Recollections", in Rossman, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya anniversary collection. Vol. 1: On Maz'ya's work in functional analysis, partial differential equations and applications. Based on talks given at the conference, Rostock, Germany, August 31 – September 4, 1998, Operator Theory. Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 1–5, ISBN 978-3-7643-6201-0, MR 1747861, Zbl 0939.01018. Hedberg, Lars Inge (1999), "On Maz'ya's work in potential theory and the theory of function spaces", in Rossmann, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya Anniversary Collection, Operator Theory: Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 7–16, doi:10.1007/978-3-0348-8675-8_2, ISBN 978-3-0348-9726-6, MR 1747862, Zbl 0939.31001 Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.
See (Hedberg 1999, pp. 10–11), (Kristensen & Mingione 2011, p. 5 and p. 8) and (Mingione 2006, p. 368). Hedberg, Lars Inge (1999), "On Maz'ya's work in potential theory and the theory of function spaces", in Rossmann, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya Anniversary Collection, Operator Theory: Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 7–16, doi:10.1007/978-3-0348-8675-8_2, ISBN 978-3-0348-9726-6, MR 1747862, Zbl 0939.31001 Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.
For more information about the work of Jindřich Nečas see the work of Kristensen & Mingione (2011, §3.3, pp. 9–12) and (Mingione 2006, §3.3, pp. 369–370). Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.

psu.edu

citeseerx.ist.psu.edu

See (Giaquinta 1983, p. 59), (Giusti 1994, p. 7 footnote 7 and p. 353), (Gohberg 1999, p. 1), (Hedberg 1999, pp. 10–11), (Kristensen & Mingione 2011, p. 5 and p. 8), and (Mingione 2006, p. 368). Giaquinta, Mariano (1983), Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton, New Jersey: Princeton University Press, pp. vii+297, ISBN 978-0-691-08330-8, MR 0717034, Zbl 0516.49003. Giusti, Enrico (1994), Metodi diretti nel calcolo delle variazioni, Monografie Matematiche (in Italian), Bologna: Unione Matematica Italiana, pp. VI+422, MR 1707291, Zbl 0942.49002, translated in English as Giusti, Enrico (2003), Direct Methods in the Calculus of Variations, River Edge, New Jersey – London – Singapore: World Scientific Publishing, pp. viii+403, doi:10.1142/9789812795557, ISBN 978-981-238-043-2, MR 1962933, Zbl 1028.49001. Gohberg, Israel (1999), "Vladimir Maz'ya: Friend and Mathematician. Recollections", in Rossman, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya anniversary collection. Vol. 1: On Maz'ya's work in functional analysis, partial differential equations and applications. Based on talks given at the conference, Rostock, Germany, August 31 – September 4, 1998, Operator Theory. Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 1–5, ISBN 978-3-7643-6201-0, MR 1747861, Zbl 0939.01018. Hedberg, Lars Inge (1999), "On Maz'ya's work in potential theory and the theory of function spaces", in Rossmann, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya Anniversary Collection, Operator Theory: Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 7–16, doi:10.1007/978-3-0348-8675-8_2, ISBN 978-3-0348-9726-6, MR 1747862, Zbl 0939.31001 Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.
See (Hedberg 1999, pp. 10–11), (Kristensen & Mingione 2011, p. 5 and p. 8) and (Mingione 2006, p. 368). Hedberg, Lars Inge (1999), "On Maz'ya's work in potential theory and the theory of function spaces", in Rossmann, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya Anniversary Collection, Operator Theory: Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 7–16, doi:10.1007/978-3-0348-8675-8_2, ISBN 978-3-0348-9726-6, MR 1747862, Zbl 0939.31001 Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.
For more information about the work of Jindřich Nečas see the work of Kristensen & Mingione (2011, §3.3, pp. 9–12) and (Mingione 2006, §3.3, pp. 369–370). Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.

semanticscholar.org

api.semanticscholar.org

See (Hilbert 1900) or, equivalently, one of its translations. Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
"Sind die Lösungen regulärer Variationsprobleme stets notwendig analytisch?" (English translation by Mary Frances Winston Newson:-"Are the solutions of regular problems in the calculus of variations always necessarily analytic?"), formulating the problem with the same words of Hilbert (1900, p. 288). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
See (Hilbert 1900, pp. 288–289), or the corresponding section on the nineteenth problem in any of its translations or reprints, or the subsection "The origins of the problem" in the historical section of this entry. Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
Hilbert does not cite explicitly Joseph Liouville and considers the constant Gaussian curvature $K$ as equal to $-1/2$ : compare the relevant entry with (Hilbert 1900, p. 288). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
Unlike Liouville's work, Picard's work is explicitly cited by Hilbert (1900, p. 288 and footnote 1 in the same page). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
See (Hilbert 1900, p. 288). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
English translation by Mary Frances Winston Newson: Hilbert's (1900, p. 288) precise words are:-"... d. h. ob jede Lagrangesche partielle Differentialgleichung eines reguläres Variationsproblem die Eigenschaft at, daß sie nur analytische Integrale zuläßt" (Italics emphasis by Hilbert himself). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
See (Giaquinta 1983, p. 59), (Giusti 1994, p. 7 footnote 7 and p. 353), (Gohberg 1999, p. 1), (Hedberg 1999, pp. 10–11), (Kristensen & Mingione 2011, p. 5 and p. 8), and (Mingione 2006, p. 368). Giaquinta, Mariano (1983), Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton, New Jersey: Princeton University Press, pp. vii+297, ISBN 978-0-691-08330-8, MR 0717034, Zbl 0516.49003. Giusti, Enrico (1994), Metodi diretti nel calcolo delle variazioni, Monografie Matematiche (in Italian), Bologna: Unione Matematica Italiana, pp. VI+422, MR 1707291, Zbl 0942.49002, translated in English as Giusti, Enrico (2003), Direct Methods in the Calculus of Variations, River Edge, New Jersey – London – Singapore: World Scientific Publishing, pp. viii+403, doi:10.1142/9789812795557, ISBN 978-981-238-043-2, MR 1962933, Zbl 1028.49001. Gohberg, Israel (1999), "Vladimir Maz'ya: Friend and Mathematician. Recollections", in Rossman, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya anniversary collection. Vol. 1: On Maz'ya's work in functional analysis, partial differential equations and applications. Based on talks given at the conference, Rostock, Germany, August 31 – September 4, 1998, Operator Theory. Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 1–5, ISBN 978-3-7643-6201-0, MR 1747861, Zbl 0939.01018. Hedberg, Lars Inge (1999), "On Maz'ya's work in potential theory and the theory of function spaces", in Rossmann, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya Anniversary Collection, Operator Theory: Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 7–16, doi:10.1007/978-3-0348-8675-8_2, ISBN 978-3-0348-9726-6, MR 1747862, Zbl 0939.31001 Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.
See (Hedberg 1999, pp. 10–11), (Kristensen & Mingione 2011, p. 5 and p. 8) and (Mingione 2006, p. 368). Hedberg, Lars Inge (1999), "On Maz'ya's work in potential theory and the theory of function spaces", in Rossmann, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya Anniversary Collection, Operator Theory: Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 7–16, doi:10.1007/978-3-0348-8675-8_2, ISBN 978-3-0348-9726-6, MR 1747862, Zbl 0939.31001 Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.
For more information about the work of Jindřich Nečas see the work of Kristensen & Mingione (2011, §3.3, pp. 9–12) and (Mingione 2006, §3.3, pp. 369–370). Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.

uni-goettingen.de

gdz.sub.uni-goettingen.de

See (Hilbert 1900) or, equivalently, one of its translations. Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
"Sind die Lösungen regulärer Variationsprobleme stets notwendig analytisch?" (English translation by Mary Frances Winston Newson:-"Are the solutions of regular problems in the calculus of variations always necessarily analytic?"), formulating the problem with the same words of Hilbert (1900, p. 288). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
See (Hilbert 1900, pp. 288–289), or the corresponding section on the nineteenth problem in any of its translations or reprints, or the subsection "The origins of the problem" in the historical section of this entry. Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
Hilbert does not cite explicitly Joseph Liouville and considers the constant Gaussian curvature $K$ as equal to $-1/2$ : compare the relevant entry with (Hilbert 1900, p. 288). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
Unlike Liouville's work, Picard's work is explicitly cited by Hilbert (1900, p. 288 and footnote 1 in the same page). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
See (Hilbert 1900, p. 288). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
English translation by Mary Frances Winston Newson: Hilbert's (1900, p. 288) precise words are:-"... d. h. ob jede Lagrangesche partielle Differentialgleichung eines reguläres Variationsproblem die Eigenschaft at, daß sie nur analytische Integrale zuläßt" (Italics emphasis by Hilbert himself). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.

web.archive.org

See (Hilbert 1900) or, equivalently, one of its translations. Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
"Sind die Lösungen regulärer Variationsprobleme stets notwendig analytisch?" (English translation by Mary Frances Winston Newson:-"Are the solutions of regular problems in the calculus of variations always necessarily analytic?"), formulating the problem with the same words of Hilbert (1900, p. 288). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
See (Hilbert 1900, pp. 288–289), or the corresponding section on the nineteenth problem in any of its translations or reprints, or the subsection "The origins of the problem" in the historical section of this entry. Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
Hilbert does not cite explicitly Joseph Liouville and considers the constant Gaussian curvature $K$ as equal to $-1/2$ : compare the relevant entry with (Hilbert 1900, p. 288). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
Unlike Liouville's work, Picard's work is explicitly cited by Hilbert (1900, p. 288 and footnote 1 in the same page). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
See (Hilbert 1900, p. 288). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
English translation by Mary Frances Winston Newson: Hilbert's (1900, p. 288) precise words are:-"... d. h. ob jede Lagrangesche partielle Differentialgleichung eines reguläres Variationsproblem die Eigenschaft at, daß sie nur analytische Integrale zuläßt" (Italics emphasis by Hilbert himself). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
See (Giaquinta 1983, p. 59), (Giusti 1994, p. 7 footnote 7 and p. 353), (Gohberg 1999, p. 1), (Hedberg 1999, pp. 10–11), (Kristensen & Mingione 2011, p. 5 and p. 8), and (Mingione 2006, p. 368). Giaquinta, Mariano (1983), Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton, New Jersey: Princeton University Press, pp. vii+297, ISBN 978-0-691-08330-8, MR 0717034, Zbl 0516.49003. Giusti, Enrico (1994), Metodi diretti nel calcolo delle variazioni, Monografie Matematiche (in Italian), Bologna: Unione Matematica Italiana, pp. VI+422, MR 1707291, Zbl 0942.49002, translated in English as Giusti, Enrico (2003), Direct Methods in the Calculus of Variations, River Edge, New Jersey – London – Singapore: World Scientific Publishing, pp. viii+403, doi:10.1142/9789812795557, ISBN 978-981-238-043-2, MR 1962933, Zbl 1028.49001. Gohberg, Israel (1999), "Vladimir Maz'ya: Friend and Mathematician. Recollections", in Rossman, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya anniversary collection. Vol. 1: On Maz'ya's work in functional analysis, partial differential equations and applications. Based on talks given at the conference, Rostock, Germany, August 31 – September 4, 1998, Operator Theory. Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 1–5, ISBN 978-3-7643-6201-0, MR 1747861, Zbl 0939.01018. Hedberg, Lars Inge (1999), "On Maz'ya's work in potential theory and the theory of function spaces", in Rossmann, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya Anniversary Collection, Operator Theory: Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 7–16, doi:10.1007/978-3-0348-8675-8_2, ISBN 978-3-0348-9726-6, MR 1747862, Zbl 0939.31001 Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.
See (Hedberg 1999, pp. 10–11), (Kristensen & Mingione 2011, p. 5 and p. 8) and (Mingione 2006, p. 368). Hedberg, Lars Inge (1999), "On Maz'ya's work in potential theory and the theory of function spaces", in Rossmann, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya Anniversary Collection, Operator Theory: Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 7–16, doi:10.1007/978-3-0348-8675-8_2, ISBN 978-3-0348-9726-6, MR 1747862, Zbl 0939.31001 Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.
For more information about the work of Jindřich Nečas see the work of Kristensen & Mingione (2011, §3.3, pp. 9–12) and (Mingione 2006, §3.3, pp. 369–370). Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.

zbmath.org

See (Hilbert 1900) or, equivalently, one of its translations. Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
"Sind die Lösungen regulärer Variationsprobleme stets notwendig analytisch?" (English translation by Mary Frances Winston Newson:-"Are the solutions of regular problems in the calculus of variations always necessarily analytic?"), formulating the problem with the same words of Hilbert (1900, p. 288). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
See (Hilbert 1900, pp. 288–289), or the corresponding section on the nineteenth problem in any of its translations or reprints, or the subsection "The origins of the problem" in the historical section of this entry. Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
Hilbert does not cite explicitly Joseph Liouville and considers the constant Gaussian curvature $K$ as equal to $-1/2$ : compare the relevant entry with (Hilbert 1900, p. 288). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
Unlike Liouville's work, Picard's work is explicitly cited by Hilbert (1900, p. 288 and footnote 1 in the same page). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
See (Hilbert 1900, p. 288). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
In his exact words: "Reguläres Variationsproblem". Hilbert's definition of a regular variational problem is stronger than the one currently used, for example, in (Gilbarg & Trudinger 2001, p. 289). Gilbarg, David; Trudinger, Neil S. (2001) [1998], Elliptic partial differential equations of second order, Classics in Mathematics (Revised 3rd printing of 2nd ed.), Berlin – Heidelberg – New York: Springer Verlag, pp. xiv+517, ISBN 978-3-540-41160-4, MR 1814364, Zbl 1042.35002.
English translation by Mary Frances Winston Newson: Hilbert's (1900, p. 288) precise words are:-"... d. h. ob jede Lagrangesche partielle Differentialgleichung eines reguläres Variationsproblem die Eigenschaft at, daß sie nur analytische Integrale zuläßt" (Italics emphasis by Hilbert himself). Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03.
– Reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German), 1: 44–63 and 253–297, 1900, JFM 32.0084.05.
– Translated to English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society, 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926.
– Reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series, 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, S2CID 12695502, Zbl 0979.01028.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques", in Duporcq, E. (ed.), Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06, archived from the original (PDF) on 2013-12-31, retrieved 2013-12-28.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French), 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
See (Giaquinta 1983, p. 59), (Giusti 1994, p. 7 footnote 7 and p. 353), (Gohberg 1999, p. 1), (Hedberg 1999, pp. 10–11), (Kristensen & Mingione 2011, p. 5 and p. 8), and (Mingione 2006, p. 368). Giaquinta, Mariano (1983), Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton, New Jersey: Princeton University Press, pp. vii+297, ISBN 978-0-691-08330-8, MR 0717034, Zbl 0516.49003. Giusti, Enrico (1994), Metodi diretti nel calcolo delle variazioni, Monografie Matematiche (in Italian), Bologna: Unione Matematica Italiana, pp. VI+422, MR 1707291, Zbl 0942.49002, translated in English as Giusti, Enrico (2003), Direct Methods in the Calculus of Variations, River Edge, New Jersey – London – Singapore: World Scientific Publishing, pp. viii+403, doi:10.1142/9789812795557, ISBN 978-981-238-043-2, MR 1962933, Zbl 1028.49001. Gohberg, Israel (1999), "Vladimir Maz'ya: Friend and Mathematician. Recollections", in Rossman, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya anniversary collection. Vol. 1: On Maz'ya's work in functional analysis, partial differential equations and applications. Based on talks given at the conference, Rostock, Germany, August 31 – September 4, 1998, Operator Theory. Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 1–5, ISBN 978-3-7643-6201-0, MR 1747861, Zbl 0939.01018. Hedberg, Lars Inge (1999), "On Maz'ya's work in potential theory and the theory of function spaces", in Rossmann, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya Anniversary Collection, Operator Theory: Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 7–16, doi:10.1007/978-3-0348-8675-8_2, ISBN 978-3-0348-9726-6, MR 1747862, Zbl 0939.31001 Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.
See (Giaquinta 1983, pp. 54–59), (Giusti 1994, p. 7 and pp. 353). Giaquinta, Mariano (1983), Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton, New Jersey: Princeton University Press, pp. vii+297, ISBN 978-0-691-08330-8, MR 0717034, Zbl 0516.49003. Giusti, Enrico (1994), Metodi diretti nel calcolo delle variazioni, Monografie Matematiche (in Italian), Bologna: Unione Matematica Italiana, pp. VI+422, MR 1707291, Zbl 0942.49002, translated in English as Giusti, Enrico (2003), Direct Methods in the Calculus of Variations, River Edge, New Jersey – London – Singapore: World Scientific Publishing, pp. viii+403, doi:10.1142/9789812795557, ISBN 978-981-238-043-2, MR 1962933, Zbl 1028.49001.
See (Hedberg 1999, pp. 10–11), (Kristensen & Mingione 2011, p. 5 and p. 8) and (Mingione 2006, p. 368). Hedberg, Lars Inge (1999), "On Maz'ya's work in potential theory and the theory of function spaces", in Rossmann, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya Anniversary Collection, Operator Theory: Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 7–16, doi:10.1007/978-3-0348-8675-8_2, ISBN 978-3-0348-9726-6, MR 1747862, Zbl 0939.31001 Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.
According to (Gohberg 1999, p. 1). Gohberg, Israel (1999), "Vladimir Maz'ya: Friend and Mathematician. Recollections", in Rossman, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.), The Maz'ya anniversary collection. Vol. 1: On Maz'ya's work in functional analysis, partial differential equations and applications. Based on talks given at the conference, Rostock, Germany, August 31 – September 4, 1998, Operator Theory. Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 1–5, ISBN 978-3-7643-6201-0, MR 1747861, Zbl 0939.01018.
See (Giaquinta 1983, pp. 54–59) and (Giusti 1994, p. 7, pp. 202–203 and pp. 317–318). Giaquinta, Mariano (1983), Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton, New Jersey: Princeton University Press, pp. vii+297, ISBN 978-0-691-08330-8, MR 0717034, Zbl 0516.49003. Giusti, Enrico (1994), Metodi diretti nel calcolo delle variazioni, Monografie Matematiche (in Italian), Bologna: Unione Matematica Italiana, pp. VI+422, MR 1707291, Zbl 0942.49002, translated in English as Giusti, Enrico (2003), Direct Methods in the Calculus of Variations, River Edge, New Jersey – London – Singapore: World Scientific Publishing, pp. viii+403, doi:10.1142/9789812795557, ISBN 978-981-238-043-2, MR 1962933, Zbl 1028.49001.
For more information about the work of Jindřich Nečas see the work of Kristensen & Mingione (2011, §3.3, pp. 9–12) and (Mingione 2006, §3.3, pp. 369–370). Kristensen, Jan; Mingione, Giuseppe (October 2011). Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas (PDF) (Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from the original (PDF) on 2014-01-07.. Mingione, Giuseppe (2006), "Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.", Applications of Mathematics, 51 (4): 355–426, CiteSeerX 10.1.1.214.9183, doi:10.1007/s10778-006-0110-3, hdl:10338.dmlcz/134645, MR 2291779, S2CID 16385131, Zbl 1164.49324.