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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hilbert’s nineteenth problem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T8850261 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Hilbert’s nineteenth problem Context triple: [Hilbert’s seventeenth problem, relatedTo, Hilbert’s nineteenth problem]
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A.
Hilbert’s twenty-second problem
Hilbert’s twenty-second problem is one of David Hilbert’s famous list of 23 problems, concerning the uniformization of analytic relations and the representation of multi-valued analytic functions by single-valued ones on suitable Riemann surfaces.
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B.
Hilbert’s twenty-third problem
Hilbert’s twenty-third problem is one of David Hilbert’s famous list of unsolved problems, focusing on the further development and systematic application of the calculus of variations.
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C.
Hilbert problems
The Hilbert problems are a famous list of 23 unsolved mathematical problems presented by David Hilbert in 1900 that profoundly influenced the development of 20th-century mathematics.
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D.
Hilbert’s seventeenth problem
Hilbert’s seventeenth problem is a famous question in real algebraic geometry asking whether every nonnegative polynomial can be represented as a sum of squares of rational functions.
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E.
Smale’s 18 problems
Smale’s 18 problems are a celebrated list of major open questions in mathematics proposed by Stephen Smale in 1998 as a successor in spirit to Hilbert’s famous problems.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Hilbert’s nineteenth problem Target entity description: 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.
-
A.
Hilbert’s twenty-second problem
Hilbert’s twenty-second problem is one of David Hilbert’s famous list of 23 problems, concerning the uniformization of analytic relations and the representation of multi-valued analytic functions by single-valued ones on suitable Riemann surfaces.
-
B.
Hilbert’s twenty-third problem
Hilbert’s twenty-third problem is one of David Hilbert’s famous list of unsolved problems, focusing on the further development and systematic application of the calculus of variations.
-
C.
Hilbert problems
The Hilbert problems are a famous list of 23 unsolved mathematical problems presented by David Hilbert in 1900 that profoundly influenced the development of 20th-century mathematics.
-
D.
Hilbert’s seventeenth problem
Hilbert’s seventeenth problem is a famous question in real algebraic geometry asking whether every nonnegative polynomial can be represented as a sum of squares of rational functions.
-
E.
Smale’s 18 problems
Smale’s 18 problems are a celebrated list of major open questions in mathematics proposed by Stephen Smale in 1998 as a successor in spirit to Hilbert’s famous problems.
- F. None of above. chosen
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
ⓘ
linked to:
Hilbert problems
|
| mainQuestion | whether solutions to regular variational problems are always analytic ⓘ |
| originalContext | foundations and future directions of mathematics ⓘ |
| partOf |
Hilbert’s list of 23 problems
ⓘ
linked to:
Hilbert problems
|
| posedBy | David Hilbert ⓘ |
| relatedTo |
Hilbert’s twentieth problem
ⓘ
elliptic regularity ⓘ variational integrals with convex integrands ⓘ |
| requiresCondition |
ellipticity of the Euler–Lagrange equations
ⓘ
sufficient smoothness of the Lagrangian ⓘ |
| solutionBy |
Ennio De Giorgi
ⓘ
John Nash ⓘ Jürgen Moser ⓘ |
| solutionDateApprox | 1950s ⓘ |
| solutionInvolves |
De Giorgi–Nash–Moser theory
ⓘ
Schauder estimates ⓘ 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
ⓘ
linked to:
Euler–Lagrange equation
elliptic systems in divergence form ⓘ |
| yearPosed | 1900 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Hilbert’s nineteenth problem Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.