Malgrange–Ehrenpreis theorem

E1021737

mathematical theorem theorem in partial differential equations

The Malgrange–Ehrenpreis theorem is a fundamental result in the theory of partial differential equations stating that every linear partial differential operator with constant coefficients admits a fundamental solution.

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Statements (46)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in partial differential equations ⓘ
appliesTo	operators on Euclidean space R^n ⓘ systems of linear PDEs with constant coefficients ⓘ
area	analysis ⓘ theory of linear operators ⓘ
assumes	coefficients of the differential operator are constant ⓘ
classification	existence theorem ⓘ
concerns	fundamental solutions of partial differential operators ⓘ linear partial differential operators with constant coefficients ⓘ
context	modern theory of linear PDEs ⓘ
doesNotRequire	ellipticity of the operator ⓘ
field	distribution theory ⓘ functional analysis ⓘ partial differential equations ⓘ
generalizes	existence of Green’s functions for ordinary differential equations with constant coefficients ⓘ
guarantees	existence of fundamental solutions for all constant coefficient linear PDEs ⓘ
hasConsequence	every constant coefficient linear PDE is locally solvable in the sense of distributions ⓘ
historicalPeriod	20th century mathematics ⓘ
implies	existence of a distribution E such that P(D)E = δ for any constant coefficient operator P(D) ⓘ
importance	provides a general method to construct solutions to linear PDEs via convolution with a fundamental solution ⓘ
influenced	development of distribution theory in PDE ⓘ subsequent work on variable coefficient operators ⓘ
involves	Dirac delta distribution ⓘ Fourier–Laplace transform NERFINISHED ⓘ polynomials in several variables ⓘ
isFundamentalResultIn	theory of linear partial differential equations with constant coefficients ⓘ
language	distribution theory ⓘ
namedAfter	Bernard Malgrange NERFINISHED ⓘ Leon Ehrenpreis NERFINISHED ⓘ
proofTechnique	Fourier analytic methods ⓘ complex analysis in several variables ⓘ
provedBy	Bernard Malgrange NERFINISHED ⓘ Leon Ehrenpreis NERFINISHED ⓘ
relatedResult	Lax–Malgrange theorem NERFINISHED ⓘ Paley–Wiener–Schwartz theorem NERFINISHED ⓘ
relatesTo	Cauchy problem for linear partial differential equations ⓘ fundamental solution method ⓘ solvability of linear PDEs ⓘ
statement	Every linear partial differential operator with constant coefficients admits a fundamental solution in the sense of distributions. ⓘ
typicalFormulation	For every nonzero polynomial P in n variables, there exists a distribution E on R^n such that P(D)E = δ_0. ⓘ
usesConcept	Fourier transform NERFINISHED ⓘ Schwartz distributions NERFINISHED ⓘ convolution ⓘ tempered distributions ⓘ
yearProvedApprox	1954 ⓘ

Referenced by (2)

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

Bernard Malgrange → hasNotableTheorem → Malgrange–Ehrenpreis theorem ⓘ

Bernard Malgrange → notableFor → Malgrange–Ehrenpreis theorem ⓘ