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 test function (smooth function with compact support) $$\varphi \in C_c^\infty(\Omega)$$, 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 $$P u$$ 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.