Hilbert’s nineteenth problem
E761265
Hilbert’s nineteenth problem is one of David Hilbert’s famous list of 23 problems, asking whether solutions to regular variational problems are always analytic.
Statements (41)
| Predicate | Object |
|---|---|
| instanceOf | Hilbert problem ⓘ |
| asksAbout |
analyticity of solutions
ⓘ
smoothness of solutions ⓘ |
| asksWhether | every minimizer of a regular variational integral is analytic ⓘ |
| assumes | regularity conditions on the integrand of the variational functional ⓘ |
| category | open problem in 1900 ⓘ |
| concerns |
analyticity of solutions of elliptic partial differential equations
ⓘ
elliptic variational problems ⓘ regularity of minimizers of variational integrals ⓘ |
| currentCategory | resolved problem in mathematics ⓘ |
| field |
calculus of variations
ⓘ
partial differential equations ⓘ regularity theory ⓘ |
| hasNumber | 19 ⓘ |
| influenced |
development of geometric measure theory
ⓘ
modern regularity theory for PDEs ⓘ nonlinear elliptic PDE theory ⓘ |
| languageOfOriginalStatement | German ⓘ |
| listedIn | Hilbert’s 1900 Paris lecture NERFINISHED ⓘ |
| mainQuestion | whether solutions to regular variational problems are always analytic ⓘ |
| originalContext | foundations and future directions of mathematics ⓘ |
| partOf | Hilbert’s list of 23 problems NERFINISHED ⓘ |
| posedBy | David Hilbert NERFINISHED ⓘ |
| relatedTo |
Hilbert’s twentieth problem
NERFINISHED
ⓘ
elliptic regularity ⓘ variational integrals with convex integrands ⓘ |
| requiresCondition |
ellipticity of the Euler–Lagrange equations
ⓘ
sufficient smoothness of the Lagrangian ⓘ |
| solutionBy |
Ennio De Giorgi
NERFINISHED
ⓘ
John Nash NERFINISHED ⓘ Jürgen Moser NERFINISHED ⓘ |
| solutionDateApprox | 1950s ⓘ |
| solutionInvolves |
De Giorgi–Nash–Moser theory
NERFINISHED
ⓘ
Schauder estimates NERFINISHED ⓘ regularity theory for elliptic partial differential equations ⓘ |
| solutionResult | under suitable regularity and ellipticity assumptions, solutions are indeed analytic ⓘ |
| status | solved ⓘ |
| typicalAssumption | integrand is sufficiently smooth and uniformly convex in the gradient variable ⓘ |
| typicalFormulationUses |
Euler–Lagrange equations
GENERATED
ⓘ
elliptic systems in divergence form GENERATED ⓘ |
| yearPosed | 1900 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.