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

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

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.