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.

References

• Gilbarg, David; Neil S. Trudinger (1988). Elliptic Partial Differential Equations of Second Order. Springer. ISBN 3-540-41160-7.
• Stein, Elias (2005). Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Princeton University Press. ISBN 0-691-11386-6.