Weyl's lemma (Laplace equation)

Template:Short description 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 $\mathrm{\Omega }$ be an open subset of $n$-dimensional Euclidean space ${ℝ}^n$, and let $\mathrm{\Delta }$ denote the usual Laplace operator. Weyl's lemma states that if a locally integrable function $u\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega })$ is a weak solution of Laplace's equation, in the sense that

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

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

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

Idea of the proof

To prove Weyl's lemma, one convolves the function $u$ with an appropriate mollifier ${\phi }_{\epsilon }$ and shows that the mollification $u_{\epsilon }={\phi }_{\epsilon }\ast u$ satisfies Laplace's equation, which implies that $u_{\epsilon }$ has the mean value property. Taking the limit as $\epsilon \to 0$ 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. Template:Math proof

Generalization to distributions

More generally, the same result holds for every distributional solution of Laplace's equation: If $T\in D^{\prime }(\mathrm{\Omega })$ satisfies $⟨T,\mathrm{\Delta }\phi ⟩=0$ for every $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$, then $T$ is a regular distribution associated with a smooth solution $u\in C^{\mathrm{\infty }}(\mathrm{\Omega })$ 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 $\mathrm{\Delta }u=0$, then the singular support of $u$ is empty since the singular support of $0$ is empty, meaning that $u\in C^{\mathrm{\infty }}(\mathrm{\Omega })$. In fact, since the Laplacian is elliptic, a stronger result is true, and solutions of $\mathrm{\Delta }u=0$ are real-analytic.

References

• Template:Cite book
• Template:Cite book