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 be an open subset of -dimensional Euclidean space , and let denote the usual Laplace operator. Weyl's lemmaHermann Weyl, The method of orthogonal projections in potential theory, Duke Math. J., 7, 411–444 (1940). See Lemma 2, p. 415 states that if a locally integrable function is a weak solution of Laplace's equation, in the sense that
for every test function (smooth function with compact support) , then (up to redefinition on a set of measure zero) is smooth and satisfies pointwise in .
This result implies the interior regularity of harmonic functions in , but it does not say anything about their regularity on the boundary .
Idea of the proof To prove Weyl's lemma, one convolves the function with an appropriate mollifier and shows that the mollification satisfies Laplace's equation, which implies that has the mean value property. Taking the limit as and using the properties of mollifiers, one finds that also has the mean value property,The mean value property is known to characterize harmonic functions in the following sense. Let . Then is harmonic in the usual sense (so and if and only if for all balls we have
where is the (n − 1)-dimensional area of the hypersphere .
Using polar coordinates about we see that when is harmonic, then for , which implies that it is a smooth solution of Laplace's equation.Bernard Dacorogna, Introduction to the Calculus of Variations, 2nd ed., Imperial College Press (2009), p. 148. Alternative proofs use the smoothness of the fundamental solution of the Laplacian or suitable a priori elliptic estimates.
Generalization to distributions More generally, the same result holds for every distributional solution of Laplace's equation: If satisfies for every , then is a regular distribution associated with a smooth solution of Laplace's equation.Lars Gårding, Some Points of Analysis and their History, AMS (1997), p. 66.
Connection with hypoellipticity Weyl's lemma follows from more general results concerning the regularity properties of elliptic or hypoelliptic operators.Lars Hörmander, The Analysis of Linear Partial Differential Operators I, 2nd ed., Springer-Verlag (1990), p.110 A linear partial differential operator with smooth coefficients is hypoelliptic if the singular support of is equal to the singular support of for every distribution . The Laplace operator is hypoelliptic, so if , then the singular support of is empty since the singular support of is empty, meaning that . In fact, since the Laplacian is elliptic, a stronger result is true, and solutions of are real-analytic.
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