Hilbert's nineteenth problem

Hilbert's nineteenth problem is one of the 23

History

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.^[4]

— David Hilbert, (Hilbert 1900, p. 288).

David Hilbert presented what is now called his nineteenth problem in his speech at the second

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.^[16] Later, Nečas (1977) provided other, more refined, examples for the vector valued problem.^[17]

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

${\displaystyle D_{i}(a^{ij}(x)\,D_{j}u)=0}$

and ${\displaystyle u}$ has square integrable first derivatives, then ${\displaystyle u}$ is Hölder continuous.

Application of De Giorgi's theorem to Hilbert's problem

Hilbert's problem asks whether the minimizers ${\displaystyle w}$ of an energy functional such as

${\displaystyle \int _{U}L(Dw)\,\mathrm {d} x}$

are analytic. Here ${\displaystyle w}$ is a function on some compact set ${\displaystyle U}$ of Rⁿ, ${\displaystyle Dw}$ is its gradient vector, and ${\displaystyle L}$ is the Lagrangian, a function of the derivatives of ${\displaystyle w}$ that satisfies certain growth, smoothness, and convexity conditions. The smoothness of ${\displaystyle w}$ can be shown using De Giorgi's theorem as follows. The Euler–Lagrange equation for this variational problem is the non-linear equation

${\displaystyle \sum \limits _{i=1}^{n}(L_{p_{i}}(Dw))_{x_{i}}=0}$

and differentiating this with respect to ${\displaystyle x_{k}}$ gives

${\displaystyle \sum \limits _{i=1}^{n}(L_{p_{i}p_{j}}(Dw)w_{x_{j}x_{k}})_{x_{i}}=0}$

This means that ${\displaystyle u=w_{x_{k}}}$ satisfies the linear equation

${\displaystyle D_{i}(a^{ij}(x)D_{j}u)=0}$

with

${\displaystyle a^{ij}=L_{p_{i}p_{j}}(Dw)}$

so by De Giorgi's result the solution w has Hölder continuous first derivatives, provided the matrix ${\displaystyle L_{p_{i}p_{j}}}$ is bounded. When this is not the case, a further step is needed: one must prove that the solution ${\displaystyle w}$ is Lipschitz continuous, i.e. the gradient ${\displaystyle Dw}$ is an ${\displaystyle L^{\infty }}$ function.

Once w is known to have Hölder continuous (n+1)st derivatives for some n ≥ 1, 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

${\displaystyle D_{i}(a^{ij}(x)D_{j}u)=D_{t}(u)}$

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

${\displaystyle D_{i}(a^{ij}(x)D_{j}u)=0}$ by considering the special case when u does not depend on t.

Notes