Hilbert's nineteenth problem

The origins of the problem

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.

David Hilbert presented the now called Hilbert's nineteen problem in his speech at the second International Congress of Mathematicians. In (Hilbert 1900, p. 288) he states that, in his opinion, one of the most remarkable facts of the theory of analytic functions is that there exist classes of partial differential equations which admit only such kind of functions as solutions, adducing Laplace's equation, Liouville's equation, the minimal surface equation and a class of linear partial differential equations studied by Émile Picard as examples. He then notes the fact that most of the partial differential equations sharing this property are the Euler–Lagrange equation of a well defined kind of variational problem, featuring the following three properties:

(1)      ∬ F ( p , q , z ; x , y ) d x d y = Minimum [ ∂ z ∂ x = p ; ∂ z ∂ y = q ] , (2)      ∂ 2 F ∂ 2 p ⋅ ∂ 2 F ∂ 2 q − ( ∂ 2 F ∂ p ∂ q ) 2 > 0 , (3)      F is an analytic function of all its arguments p, q, z, x and y.

Hilbert calls this kind of variational problem a "regular variational problem": property (1) means that such kind of variational problems are minimum problems, property (2) is the ellipticity condition on the Euler–Lagrange equations associated to the given functional, while property (3) is a simple regularity assumption the function F. Having identified the class of problems to deal with, he then poses the following question:-"... does every Lagrangian partial differential equation of a regular variation problem have the property of admitting analytic integrals exclusively?" and asks further if this is the case even when the function is required to assume, as it happens for Dirichlet's problem on the potential function, boundary values which are continuous, but not analytic.

The path to the complete solution

Hilbert stated his nineteenth problem as a regularity problem for a class of elliptic partial differential equation with analytic coefficients, therefore the first efforts of the researchers who sought to solve it were directed to study 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: he showed that C 3  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.

Counterexamples to various generalizations of the problem

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 sixties, Maz'ya (1968), De Giorgi (1968) and Giusti & Miranda (1968) constructed independently several counterexamples, showing that in general there is no hope to prove such kind of 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: for experts, the fact that such kind of equations could have nonanalytic and even nonsmooth solutions created a sensation.

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. Later on, Nečas (1977) provided other, more refined, examples for the vector valued problem.

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

D i ( a i j ( x ) D j u ) = 0

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

∫ U L ( D w ) d x

are analytic. Here w is a function on some compact set U of Rⁿ, D w is its gradient vector, 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

Σ i ( L p i ( D w ) ) x i = 0

and differentiating this with respect to x_k gives

Σ i ( L p i p j ( D w ) w x j x k ) x i = 0

This means that u=w_xk satisfies the linear equation

D i ( a i j ( x ) D j u ) = 0

with

a i j = L p i p j ( D w )

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 a^ij 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

D i ( a i j ( x ) D j u ) = D t ( u )

where u is a bounded function of x₁,...,x_n, t defined for t ≥ 0. From his estimate Nash was able to deduce a continuity estimate for solutions of the elliptic equation

D i ( a i j ( x ) D j u ) = 0 by considering the special case when u does not depend on t.