Weyl's lemma (Laplace equation)

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 $\Omega$ be an open subset of $n$-dimensional Euclidean space $\mathbb{R}^{n}$, and let $\Delta$ denote the usual Laplace operator. Weyl's lemma states that if a locally integrable function $u \in L_{\mathrm{loc}}^{1}(\Omega)$ is a weak solution of Laplace's equation, in the sense that

$\int_{\Omega} u(x) \Delta \phi (x) \, dx = 0$

for every smooth test function $\phi \in C_c^\infty(\Omega)$ with compact support, then (up to redefinition on a set of measure zero) $u \in C^{\infty}(\Omega)$ is smooth and satisfies $\Delta u = 0$ pointwise in $\Omega$.

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.

Generalization to distributions

More generally, the same result holds for every distributional solution of Laplace's equation: If $T\in D'(\Omega)$ satisfies $\langle T, \Delta \phi\rangle = 0$ for every $\phi\in C_c^\infty(\Omega)$, then $T= T_u$ is a regular distribution associated with a smooth solution $u\in C^\infty(\Omega)$ of Laplace's equation.

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 $P u$ is equal to the singular support of $u$ for every distribution $u$. The Laplace operator is hypoelliptic, 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.