Hilbert's nineteenth problem
Hilbert's nineteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It asks whether the solutions of regular problems in the calculus of variations are always analytic.
For C3 solutions Hilbert's problem was answered positively by Sergei Bernstein (1904) in his thesis, who showed that C3 solutions of nonlinear elliptic analytic equations in 2 variables are analytic. Bernstein's result was improved over the years by several authors, such as Petrowsky (1939), who reduced the differentiability requirements on the solution needed to prove that it is analytic. On the other hand direct methods in the calculus of variations showed the existence of solutions with very weak differentiability properties. For many years there was a gap between these results: the solutions that could be constructed were known to have square integrable second derivatives, which was not quite strong enough to feed into the machinery that could prove they were analytic, which needed continuity of first derivatives. This gap was filled independently by Ennio De Giorgi (1956, 1957), and John Forbes Nash (1957, 1958). They were able to show the solutions had first derivatives that were Hölder continuous, which by previous results implied that the solutions are analytic whenever the differential equation has analytic coefficients, thus completing the solution of Hilbert's nineteenth problem.
De Giorgi (1968) gave a counterexample showing that in the case when the solution is vector-valued rather than scalar-valued, the solution need not be analytic.
De Giorgi's theorem 
The key theorem proved by De Giorgi is an a priori estimate stating that if u is a solution of a suitable linear second order strictly elliptic PDE of the form
and u has square integrable first derivatives, then u is Hölder continuous.
Application of De Giorgi's theorem to Hilbert's problem 
Hilbert's problem asks whether the minimizers w of an energy functional such as
are analytic. Here w is a function on some compact set U of Rn, Dw is the vector of its first derivatives, and L is the Lagrangian, a function of the derivatives of w that satisfies certain growth, smoothness, and convexity conditions. The smoothness of w can be shown using De Giorgi's theorem as follows. The Euler-Lagrange equation for this variational problem is the non-linear equation
and differentiating this with respect to xk gives
This means that u=wxk satisfies the linear equation
so by De Giorgi's result the solution w has Hölder continuous first derivatives.
Once w is known to have Hölder continuous (n+1)st derivatives for some n ≥ 0, then the coefficients aij have Hölder continuous nth derivatives, so a theorem of Schauder implies that the (n+2)nd derivatives are also Hölder continuous, so repeating this infinitely often shows that the solution w is smooth.
Nash's theorem 
Nash gave a continuity estimate for solutions of the parabolic equation
where u is a bounded function of x1,...,xn, t defined for t ≥ 0. From his estimate Nash was able to deduce a continuity estimate for solutions of the elliptic equation
- by considering the special case when u does not depend on t.
- Bernstein, S. (1904), "Sur la nature analytique des solutions des équations aux dérivées partielles du second ordre", Mathematische Annalen (Springer Berlin / Heidelberg) 59: 20–76, doi:10.1007/BF01444746, ISSN 0025-5831
- Bombieri, Enrico (1975), "Variational problems and elliptic equations", Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974), Vol. 1, Canad. Math. Congress, Montreal, Que., pp. 53–63, MR 0509259 Reprinted in Bombieri, Enrico (1976), "Variational problems and elliptic equations", in Browder, Felix E., Mathematical developments arising from Hilbert problems (Proceedings of the Symposium in Pure Mathematics of the American Mathematical Society held at Northern Illinois University, De Kalb, Ill., May, 1974. ), Proc. Sympos. Pure Math., XXVIII, Providence, R.I.: Amer. Math. Soc., pp. 525–535, ISBN 978-0-8218-1428-4, MR 0425740
- De Giorgi, Ennio (1956), "Sull'analiticità delle estremali degli integrali multipli", Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali (in Italian) 20: 438–441, MR 0082045 English translation in (De Giorgi 2006)
- De Giorgi, Ennio (1957), "Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari", Memorie della Accademia delle Scienze di Torino. Classe di Scienze Fisiche, Matematicahe e Naturali. (3) (in Italian) 3: 25–43, MR 0093649 English translation in (De Giorgi 2006)
- De Giorgi, Ennio (1968), "Un esempio di estremali discontinue per un problema variazionale di tipo ellittico", Boll. Un. Mat. Ital. (4) (in Italian) 1: 135–137, MR 0227827 English translation in (De Giorgi 2006)
- De Giorgi, Ennio (2006), in Ambrosio, Luigi; Maso, Gianni Dal; Forti, Marco; Miranda, Mario; Spagnolo, Sergio, Selected papers, Berlin, New York: Springer-Verlag, ISBN 978-3-540-26169-8, MR 2229237
- Nash, John (1957), "Parabolic equations", Proceedings of the National Academy of Sciences of the United States of America 43: 754–758, ISSN 0027-8424, JSTOR 89599, MR 0089986
- Nash, John (1958), "Continuity of solutions of parabolic and elliptic equations", American Journal of Mathematics 80: 931–954, ISSN 0002-9327, JSTOR 2372841, MR 0100158
- Petrowsky, I. G. (1939), "Sur l'analyticité des solutions des systèmes d'équations différentielles", Rec. Math. (Mat. Sbornik) N.S., 5(47) (1): 3–70, MR 0001425