Orthogonal convex hull

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In mathematics, Weyl's lemma, named after Hermann Weyl, states that every weak solution of Laplace's equation is a smooth solution. This contrasts with the wave equation, for example, which has weak solutions that are not smooth solutions. Weyl's lemma is a special case of elliptic or hypoelliptic regularity.

Statement of the lemma

Let Ω be an open subset of n-dimensional Euclidean space n, and let Δ denote the usual Laplace operator. Weyl's lemma[1] states that if a locally integrable function uLloc1(Ω) is a weak solution of Laplace's equation, in the sense that

Ωu(x)Δϕ(x)dx=0

for every smooth test function ϕCc(Ω) with compact support, then (up to redefinition on a set of measure zero) uC(Ω) is smooth and satisfies Δu=0 pointwise in Ω.

This result implies the interior regularity of harmonic functions in Ω, but it does not say anything about their regularity on the boundary Ω.

Idea of the proof

To prove Weyl's lemma, one convolves the function u with an appropriate mollifier ϕϵ and shows that the mollification uϵ=ϕϵu satisfies Laplace's equation, which implies that uϵ has the mean value property. Taking the limit as ϵ0 and using the properties of mollifiers, one finds that u also has the mean value property, which implies that it is a smooth solution of Laplace's equation.[2]

Generalization to distributions

More generally, the same result holds for every distributional solution of Laplace's equation: If TD(Ω) satisfies T,Δϕ=0 for every ϕCc(Ω), then T=Tu is a regular distribution associated with a smooth solution uC(Ω) of Laplace's equation.[3]

Connection with hypoellipticity

Weyl's lemma follows from more general results concerning the regularity properties of elliptic or hypoelliptic operators.[4] A linear partial differential operator P with smooth coefficients is hypoelliptic if the singular support of Pu is equal to the singular support of u for every distribution u. The Laplace operator is hyperelliptic, so if Δu=0, then the singular support of u is empty since the singular support of 0 is empty, meaning that uC(Ω). In fact, since the Laplacian is elliptic, a stronger result is true, and solutions of Δu=0 are real-analytic.

Notes

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References

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  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  1. Hermann Weyl, The method of orthogonal projections in potential theory, Duke Math. J., 7, 411-444 (1940). See Lemma 2, p. 415
  2. Bernard Daconorogna, Introduction to the Calculus of Variations, 2nd ed., Imperial College Press (2009), p. 148.
  3. Lars Gårding, Some Points of Analysis and their History, AMS (1997), p. 66.
  4. Lars Hörmander, The Analysis of Linear Partial Differential Operators I, 2nd ed., Springer-Verlag (1990), p.110