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| 19th problem of Hilbert | |
|---|---|
| Name | 19th problem of Hilbert |
| Field | Calculus of Variations |
| Proposer | David Hilbert |
| Year | 1900 |
| Status | Solved |
19th problem of Hilbert
The 19th problem of Hilbert asked whether solutions of elliptic variational problems are necessarily analytic. It sits at the intersection of David Hilbert, Hilbert's problems, calculus of variations, elliptic partial differential equations, and regularity theory and motivated developments across Bernhard Riemann, Carl Friedrich Gauss, Émile Picard, Jean le Rond d'Alembert, and later analysts such as Ennio De Giorgi, John Nash, and Giovanni Maria Cerami. The question prompted work linking Karl Weierstrass, Henri Lebesgue, Bernhard Riemann's successors, and institutions like the University of Göttingen and the École Normale Supérieure.
Hilbert posed whether minimizers of a regular variational integral with analytic integrand must be analytic solutions of the associated Euler–Lagrange equation. The formulation referenced classical sources including Leonhard Euler, Joseph-Louis Lagrange, Adrien-Marie Legendre, and sought implications for problems studied by Bernhard Riemann and Gustav Kirchhoff. Hilbert's statement was embedded among his list presented at the International Congress of Mathematicians in Paris, and it invoked the analytic regularity results anticipated by proponents such as Henri Poincaré and Émile Picard.
The problem emerged from 19th-century analysis and mechanics where figures like Carl Friedrich Gauss, Sofia Kovalevskaya, and George Gabriel Stokes investigated smoothness of solutions. Hilbert framed it during the rise of formal analysis at the University of Göttingen and in response to variational work by Lord Kelvin, William Rowan Hamilton, and Augustin-Louis Cauchy. The question stimulated research programs at centers including Princeton University, University of Cambridge, and Scuola Normale Superiore di Pisa, and engaged mathematicians from schools associated with Felix Klein, Emmy Noether, and Hermann Weyl.
Progress toward resolution came in stages: early partial results by Jacques Hadamard and Émile Borel addressed special integrands; counterexamples by Karl Weierstrass highlighted pathologies. The breakthrough regularity theory for elliptic minima was developed by Ennio De Giorgi, John Nash, and Sergei Sobolev, culminating in proofs that under natural convexity and smoothness hypotheses minimizers are analytic. Subsequent refinements were achieved by Laurent Caffarelli, Luis Caffarelli, Eberhard Hopf, and Kurt Friedrichs expanding links with Sobolev spaces and the Dirichlet problem. Important concepts from Sergio Bernstein and Lars Ahlfors influenced boundary regularity results.
Techniques combined classical calculus of variations from Leonhard Euler and Joseph-Louis Lagrange with modern functional analysis inspired by Stefan Banach, John von Neumann, and Andrey Kolmogorov. Key tools included the method of continuity used by Paul Garabedian, energy estimates from Marcel Riesz, compactness via Vitali, and embedding theorems related to Sergei Sobolev and Norbert Wiener. De Giorgi's measure-theoretic approach intertwined with Nash's a priori estimates, while elliptic regularity used Schauder theory linked to Julian Schauder and Calderón–Zygmund theory associated with Alberto Calderón and Antoni Zygmund.
Hilbert's 19th motivated extensions such as regularity for quasi-linear systems studied by James Serrin and Mikhail Lavrentyev, and influenced the study of minimal surfaces by Jesse Douglas and Ennio De Giorgi's contemporaries. It connected to the 20th problem about existence posed by Hilbert and to modern questions in geometric analysis pursued by Michael Atiyah, Isadore Singer, and Shing-Tung Yau. Further related problems include regularity in nonlinear elasticity examined by John Ball and regularity for degenerate elliptic equations considered by Oleksandr Oleinik.
Seminal contributions include De Giorgi's papers establishing Hölder continuity for elliptic equations, Nash's independent regularity proofs, and foundational work by Sergei Sobolev on function spaces. Major expositions appear in works by Eberhard Hopf, J. J. Stoker, Leon Simon, and survey treatments by Lawrence C. Evans and Michael Taylor. Historical analyses and modern syntheses were advanced at seminars in Princeton University and the Institut des Hautes Études Scientifiques, while influential journals such as Acta Mathematica, Annals of Mathematics, and Inventiones Mathematicae disseminated the pivotal papers.
Category:Hilbert problems