Hilbert's nineteenth problem was solved independently in the late 1950s by Ennio De Giorgi and John Forbes Nash, Jr. Eine der begrifflich merkwürdigsten Thatsachen in den Elementen der Theorie der analytischen Funktionen erblicke ich darin, daß es Partielle Differentialgleichungen giebt, deren Integrale sämtlich notwendig analytische Funktionen der unabhängigen Variabeln sind, die also, kurz gesagt, nur analytischer Lösungen fähig sind.
[4]David Hilbert presented what is now called his nineteenth problem in his speech at the second International Congress of Mathematicians.
[7] He then notes that most partial differential equations sharing this property are Euler–Lagrange equations of a well defined kind of variational problem, satisfying the following three properties:[8] Hilbert calls this a "regular variational problem".
[8] Therefore the first efforts of researchers who sought to solve it were aimed at studying the regularity of classical solutions for equations belonging to this class.
For C 3 solutions, Hilbert's problem was answered positively by Sergei Bernstein (1904) in his thesis.
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.
This gap was filled independently by Ennio De Giorgi (1956, 1957), and John Forbes Nash (1957, 1958), who were able to show the solutions had first derivatives that were Hölder continuous.
Subsequently, Jürgen Moser gave an alternate proof of the results obtained by Ennio De Giorgi (1956, 1957), and John Forbes Nash (1957, 1958).
The affirmative answer to Hilbert's nineteenth problem given by Ennio De Giorgi and John Forbes Nash raised the question if the same conclusion holds also for Euler–Lagrange equations of more general functionals.
At the end of the 1960s, Maz'ya (1968),[12] De Giorgi (1968) and Giusti & Miranda (1968) independently constructed several counterexamples,[13] showing that in general there is no hope of proving such regularity results without adding further hypotheses.
Precisely, Maz'ya (1968) gave several counterexamples involving a single elliptic equation of order greater than two with analytic coefficients.
[14] For experts, the fact that such equations could have nonanalytic and even nonsmooth solutions created a sensation.
[15] De Giorgi (1968) and Giusti & Miranda (1968) gave counterexamples showing that in the case when the solution is vector-valued rather than scalar-valued, it need not be analytic; the example of De Giorgi consists of an elliptic system with bounded coefficients, while the one of Giusti and Miranda has analytic coefficients.
[17] 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
satisfies the linear equation with so by De Giorgi's result the solution w has Hölder continuous first derivatives, provided the matrix
John 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.