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Partial differential equations Qualitative behavior and nonlinear analysis Problem

Hilbert's nineteenth problem

Builds on Calculus of variationsEuler–Lagrange equationAnalytic function

Problem

Let Ω⊆ℝn\Omega \subseteq \R^n be an open set and let L ⁣:ℝn→ℝL \colon \R^n \to \R, written L(p)L(p), be a real analytic function. Assume LL is uniformly convex: for constants 0<λ≤Λ0 < \lambda \le \Lambda, the eigenvalues of the Hessian matrix D2L(p)D^2 L(p) lie in [λ,Λ][\lambda, \Lambda] for all pp. Consider the functional J(u)=∫ΩL(∇u) dxJ(u) = \int_\Omega L(\nabla u) \, dx of the calculus of variations. Call u∈Wloc1,2(Ω)u \in W^{1,2}_{\mathrm{loc}}(\Omega) a local minimiser of JJ if ∫SL(∇u) dx≤∫SL(∇u+∇φ) dx\int_{S} L(\nabla u) \, dx \le \int_{S} L(\nabla u + \nabla \varphi) \, dx for every φ∈W1,2(Ω)\varphi \in W^{1,2}(\Omega) that vanishes outside a compact set S⊆ΩS \subseteq \Omega. Decide whether every local minimiser of JJ is real analytic in Ω\Omega. In Hilbert's words, the question is whether the solutions of regular problems of the calculus of variations are always analytic.

Remarks

Hilbert asked the question in a more general form, for analytic Lagrangians L(x,u,∇u)L(x, u, \nabla u) that satisfy the Legendre condition; the form above is the model case, with the same answer. The answer is yes for scalar functions uu. Bernstein proved in 1904 that C3C^3 solutions of analytic nonlinear elliptic equations in two variables are analytic. The general case was settled by De Giorgi in 1957 and, independently, by Nash in 1958. They proved that weak solutions of linear elliptic equations in divergence form with bounded measurable coefficients are Hölder continuous. The derivatives of a minimiser solve such an equation, so u∈C1,αu \in C^{1,\alpha}. Then Schauder theory gives u∈C∞u \in C^\infty, and the analyticity of smooth solutions of analytic elliptic equations gives that uu is analytic. For vector-valued uu the answer is no: De Giorgi gave a counterexample in 1968 of a non-smooth minimiser for an elliptic system.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Hilbert%27s_nineteenth_problem
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Changed2 November 2026

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