# Weyl's lemma (Laplace equation)

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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

$\int _{\Omega }u(x)\Delta \phi (x)\,dx=0$ This result implies the interior regularity of harmonic functions in $\Omega$ , but it does not say anything about their regularity on the boundary $\partial \Omega$ .

## Idea of the proof

To prove Weyl's lemma, one convolves the function $u$ with an appropriate mollifier $\phi _{\epsilon }$ and shows that the mollification $u_{\epsilon }=\phi _{\epsilon }\ast u$ satisfies Laplace's equation, which implies that $u_{\epsilon }$ has the mean value property. Taking the limit as $\epsilon \to 0$ 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. Alternative proofs use the smoothness of the fundamental solution of the Laplacian or suitable a priori elliptic estimates.

## Connection with hypoellipticity

Weyl's lemma follows from more general results concerning the regularity properties of elliptic or hypoelliptic operators. 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 $\Delta u=0$ , then the singular support of $u$ is empty since the singular support of $0$ is empty, meaning that $u\in C^{\infty }(\Omega )$ . In fact, since the Laplacian is elliptic, a stronger result is true, and solutions of $\Delta u=0$ are real-analytic.