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

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

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

This result implies the interior regularity of harmonic functions in ${\displaystyle \Omega }$, but it does not say anything about their regularity on the boundary ${\displaystyle \partial \Omega }$.

## Idea of the proof

To prove Weyl's lemma, one convolves the function ${\displaystyle u}$ with an appropriate mollifier ${\displaystyle \phi _{\epsilon }}$ and shows that the mollification ${\displaystyle u_{\epsilon }=\phi _{\epsilon }\ast u}$ satisfies Laplace's equation, which implies that ${\displaystyle u_{\epsilon }}$ has the mean value property. Taking the limit as ${\displaystyle \epsilon \to 0}$ and using the properties of mollifiers, one finds that ${\displaystyle u}$ also has the mean value property, which implies that it is a smooth solution of Laplace's equation.[2] 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 ${\displaystyle T\in D'(\Omega )}$ satisfies ${\displaystyle \langle T,\Delta \phi \rangle =0}$ for every ${\displaystyle \phi \in C_{c}^{\infty }(\Omega )}$, then ${\displaystyle T=T_{u}}$ is a regular distribution associated with a smooth solution ${\displaystyle u\in C^{\infty }(\Omega )}$ 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 ${\displaystyle P}$ with smooth coefficients is hypoelliptic if the singular support of ${\displaystyle Pu}$ is equal to the singular support of ${\displaystyle u}$ for every distribution ${\displaystyle u}$. The Laplace operator is hypoelliptic, so if ${\displaystyle \Delta u=0}$, then the singular support of ${\displaystyle u}$ is empty since the singular support of ${\displaystyle 0}$ is empty, meaning that ${\displaystyle u\in C^{\infty }(\Omega )}$. In fact, since the Laplacian is elliptic, a stronger result is true, and solutions of ${\displaystyle \Delta u=0}$ are real-analytic.

## Notes

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 Dacorogna, 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