# 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

$\int _{\Omega }u(x)\,\Delta \varphi (x)\,dx=0$

for every smooth test function $\varphi \in C_{c}^{\infty }(\Omega )$  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 $\varphi _{\varepsilon }$  and shows that the mollification $u_{\varepsilon }=\varphi _{\varepsilon }\ast u$  satisfies Laplace's equation, which implies that $u_{\varepsilon }$  has the mean value property. Taking the limit as $\varepsilon \to 0$  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 $\langle T,\Delta \varphi \rangle =0$  for every $\varphi \in C_{c}^{\infty }(\Omega )$ , 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 $Pu$  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.