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 Ω {\displaystyle \Omega } be an open subset of n {\displaystyle n} -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} , and let Δ {\displaystyle \Delta } denote the usual Laplace operator. Weyl's lemma states that if a locally integrable function u ∈ L l o c 1 ( Ω ) {\displaystyle u\in L_{\mathrm {loc} }^{1}(\Omega )} is a weak solution of Laplace's equation, in the sense that
∫ Ω u ( x ) Δ φ ( x ) d x = 0 {\displaystyle \int _{\Omega }u(x)\,\Delta \varphi (x)\,dx=0}
for every test function (smooth function with compact support) φ ∈ C c ∞ ( Ω ) {\displaystyle \varphi \in C_{c}^{\infty }(\Omega )} , then (up to redefinition on a set of measure zero) u ∈ C ∞ ( Ω ) {\displaystyle u\in C^{\infty }(\Omega )} is smooth and satisfies Δ u = 0 {\displaystyle \Delta u=0} pointwise in Ω {\displaystyle \Omega } . This result implies the interior regularity of harmonic functions in Ω {\displaystyle \Omega } , but it does not say anything about their regularity on the boundary ∂ Ω {\displaystyle \partial \Omega } .
Idea of the proof To prove Weyl's lemma, one convolves the function u {\displaystyle u} with an appropriate mollifier φ ε {\displaystyle \varphi _{\varepsilon }} and shows that the mollification u ε = φ ε ∗ u {\displaystyle u_{\varepsilon }=\varphi _{\varepsilon }\ast u} satisfies Laplace's equation, which implies that u ε {\displaystyle u_{\varepsilon }} has the mean value property. Taking the limit as ε → 0 {\displaystyle \varepsilon \to 0} and using the properties of mollifiers, one finds that u {\displaystyle 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 ∈ D ′ ( Ω ) {\displaystyle T\in D'(\Omega )} satisfies ⟨ T , Δ φ ⟩ = 0 {\displaystyle \langle T,\Delta \varphi \rangle =0} for every φ ∈ C c ∞ ( Ω ) {\displaystyle \varphi \in C_{c}^{\infty }(\Omega )} , then T {\displaystyle T} is a regular distribution associated with a smooth solution u ∈ C ∞ ( Ω ) {\displaystyle 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 {\displaystyle P} with smooth coefficients is hypoelliptic if the singular support of P u {\displaystyle Pu} is equal to the singular support of u {\displaystyle u} for every distribution u {\displaystyle u} . The Laplace operator is hypoelliptic, so if Δ u = 0 {\displaystyle \Delta u=0} , then the singular support of u {\displaystyle u} is empty since the singular support of 0 {\displaystyle 0} is empty, meaning that u ∈ C ∞ ( Ω ) {\displaystyle u\in C^{\infty }(\Omega )} . In fact, since the Laplacian is elliptic, a stronger result is true, and solutions of Δ u = 0 {\displaystyle \Delta u=0} are real-analytic.
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