Dirichlet principle
E466246
The Dirichlet principle is a foundational concept in potential theory and the calculus of variations that asserts certain boundary value problems can be solved by finding a function minimizing an associated energy integral.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Dirichlet principle canonical | 1 |
| pigeonhole principle | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4746234 — 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: Dirichlet principle Context triple: [Peter Gustav Lejeune Dirichlet, notableWork, Dirichlet principle]
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A.
de Bruijn–van Aardenne–Ehrenfest theorem
The de Bruijn–van Aardenne–Ehrenfest theorem is a fundamental result in combinatorics that characterizes the number of Eulerian circuits in directed graphs, particularly de Bruijn graphs, and underpins constructions in coding theory and discrete mathematics.
-
B.
Euler’s theorem
Euler’s theorem is a fundamental result in number theory stating that for any integer a coprime to n, a raised to the power of φ(n) is congruent to 1 modulo n.
-
C.
Sperner's lemma
Sperner's lemma is a fundamental result in combinatorial topology that guarantees the existence of a fully labeled simplex in certain labeled triangulations, and is widely used to prove fixed-point and equilibrium theorems.
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D.
Mertens’ theorems
Mertens’ theorems are classical results in analytic number theory that give precise asymptotic estimates for sums involving the Möbius function and the reciprocals of primes, illuminating the distribution of primes and their connection to the Riemann zeta function.
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E.
Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
- 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: Dirichlet principle Target entity description: The Dirichlet principle is a foundational concept in potential theory and the calculus of variations that asserts certain boundary value problems can be solved by finding a function minimizing an associated energy integral.
-
A.
de Bruijn–van Aardenne–Ehrenfest theorem
The de Bruijn–van Aardenne–Ehrenfest theorem is a fundamental result in combinatorics that characterizes the number of Eulerian circuits in directed graphs, particularly de Bruijn graphs, and underpins constructions in coding theory and discrete mathematics.
-
B.
Euler’s theorem
Euler’s theorem is a fundamental result in number theory stating that for any integer a coprime to n, a raised to the power of φ(n) is congruent to 1 modulo n.
-
C.
Sperner's lemma
Sperner's lemma is a fundamental result in combinatorial topology that guarantees the existence of a fully labeled simplex in certain labeled triangulations, and is widely used to prove fixed-point and equilibrium theorems.
-
D.
Mertens’ theorems
Mertens’ theorems are classical results in analytic number theory that give precise asymptotic estimates for sums involving the Möbius function and the reciprocals of primes, illuminating the distribution of primes and their connection to the Riemann zeta function.
-
E.
Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
existence principle
ⓘ
mathematical principle ⓘ |
| appearsIn |
classical potential theory textbooks
ⓘ
treatises on elliptic partial differential equations ⓘ |
| appliesTo |
Dirichlet problem
NERFINISHED
ⓘ
Laplace equation NERFINISHED ⓘ boundary value problems ⓘ harmonic functions ⓘ |
| assumes | existence of a function that minimizes the Dirichlet integral ⓘ |
| boundaryConditionType | Dirichlet boundary condition NERFINISHED ⓘ |
| coreIdea |
a function with prescribed boundary values that minimizes the Dirichlet energy is harmonic inside the domain
ⓘ
solutions to certain boundary value problems can be obtained as minimizers of an energy integral ⓘ |
| criticizedBy | Karl Weierstrass NERFINISHED ⓘ |
| field |
calculus of variations
ⓘ
mathematical analysis ⓘ partial differential equations ⓘ potential theory ⓘ |
| formalization | can be formulated as a minimization problem in appropriate function spaces ⓘ |
| historicalStatus |
its original form was criticized for lack of rigor
ⓘ
was originally used heuristically by Dirichlet ⓘ |
| influenced |
axiomatization of Hilbert spaces
ⓘ
development of functional analysis ⓘ development of modern calculus of variations ⓘ |
| involvesConcept |
Dirichlet energy
NERFINISHED
ⓘ
Sobolev spaces NERFINISHED ⓘ elliptic partial differential equations ⓘ energy integral ⓘ harmonic function ⓘ minimization of functionals ⓘ variational problem ⓘ weak solutions ⓘ |
| logicalForm | existence statement for minimizers of an energy functional ⓘ |
| madeRigorousBy |
David Hilbert
NERFINISHED
ⓘ
Henri Poincaré NERFINISHED ⓘ |
| namedAfter | Peter Gustav Lejeune Dirichlet NERFINISHED ⓘ |
| relatedTo |
Green function
ⓘ
Hilbert space methods ⓘ Riesz representation theorem NERFINISHED ⓘ direct method in the calculus of variations ⓘ maximum principle for harmonic functions ⓘ |
| requiresCondition |
coercivity of the energy functional
ⓘ
lower semicontinuity of the energy functional ⓘ |
| status | accepted as rigorous when formulated in modern functional analytic terms ⓘ |
| timePeriod | 19th century ⓘ |
| typicalDomain | bounded domain in Euclidean space ⓘ |
| usedFor |
constructing Green functions in potential theory
ⓘ
establishing existence of solutions to elliptic boundary value problems ⓘ proving existence of harmonic functions with prescribed boundary values ⓘ |
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: Dirichlet principle Description of subject: The Dirichlet principle is a foundational concept in potential theory and the calculus of variations that asserts certain boundary value problems can be solved by finding a function minimizing an associated energy integral.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
pigeonhole principle