Dirichlet principle

E466246

existence principle mathematical principle

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.

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All labels observed (2)

Label	Occurrences
Dirichlet principle canonical	1
pigeonhole principle	1

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 ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Peter Gustav Lejeune Dirichlet → notableWork → Dirichlet principle ⓘ

Ramsey theory → basedOn → Dirichlet principle ⓘ

this entity surface form: pigeonhole principle