Hilberts nittonde problem

Hilberts nittonde problem är ett av Hilberts 23 problem. Det formulerades år 1900 relaterat till frågan:

Är lösningarna till Lagranges ekvationer alltid analytiska?

Matematikern John Forbes Nash bevisade på 1950-talet att svaret på frågan är "ja": lösningarna är alltid analytiska.

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Den här artikeln är helt eller delvis baserad på material från engelskspråkiga Wikipedia, Hilbert's nineteenth problem, 8 januari 2014.

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• Gohberg, Israel (1999), ”Vladimir Maz'ya: Friend and Mathematician. Recollections”, i Rossman, Jürgen; Takáč, Peter; Wildenhain, Günther, 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, "109", Basel: Birkhäuser Verlag, s. 1–5, ISBN 978-3-7643-6201-0, http://books.google.com/?id=9xPz9Mg2c_EC&printsec=frontcover#v=onepage&q .
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• Kristensen, Jan; Mingione, Giuseppe (October 2011), Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas, "Report no. OxPDE-11/17", Oxford: Oxford Centre for Nonlinear PDE, s. 1–30, arkiverad från ursprungsadressen den 2014-01-07, https://web.archive.org/web/20140107114055/http://www.maths.ox.ac.uk/system/files/attachments/OxPDE_11-17.pdf .
• Maz'ya, V. G. (1968), ”Примеры нерегулярных решений квазилинейных эллиптических уравнений с аналитическими коэффициентами” (på russian), Funktsional’nyĭ Analiz i Ego Prilozheniya 2 (3): 53–57, http://mi.mathnet.ru/eng/faa/v2/i3/p53 , translated in English as Maz'ya, V. G. (1968), ”Examples of nonregular solutions of quasilinear elliptic equations with analytic coefficients”, Functional Analysis and Its Applications 2 (3): 230-234, doi:10.1007/BF01076124 .
• Mingione, Giuseppe (2006), ”Regularity of minima: an invitation to the Dark Side of the Calculus of Variations.”, Applications of Mathematics 51 (4): 355–426, http://dml.cz/dmlcz/134645 .
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• Nash, John (1957), ”Parabolic equations”, Proceedings of the National Academy of Sciences of the United States of America 43 (8): 754–758, ISSN 0027-8424, http://www.pnas.org/content/43/8/754.full.pdf+html?sid=db030833-a739-437a-8ce0-be81f750b3a7 .
• Nash, John (1958), ”Continuity of solutions of parabolic and elliptic equations”, American Journal of Mathematics 80 (4): 931–954, ISSN 0002-9327 .
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• Petrowsky, I. G. (1939), ”Sur l'analyticité des solutions des systèmes d'équations différentielles” (på franska), Recueil Mathématique (Matematicheskii Sbornik) 5(47) (1): 3–70, http://mi.mathnet.ru/eng/msb5769 .