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

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

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

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.

This page was last edited on 25 April 2023, at 18:08