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.
Let Ω be an open subset of n -dimensional Euclidean space R n , and let Δ denote the usual Laplace operator. Weyl's lemma states that if a locally integrable function u ∈ L l o c 1 ( Ω ) is a weak solution of Laplace's equation, in the sense that
∫ Ω u ( x ) Δ ϕ ( x ) d x = 0 for every smooth test function ϕ ∈ C c ∞ ( Ω ) with compact support, then (up to redefinition on a set of measure zero) u ∈ C ∞ ( Ω ) is smooth and satisfies Δ u = 0 pointwise in Ω .
This result implies the interior regularity of harmonic functions in Ω , but it does not say anything about their regularity on the boundary ∂ Ω .
To prove Weyl's lemma, one convolves the function u with an appropriate mollifier ϕ ϵ and shows that the mollification u ϵ = ϕ ϵ ∗ u satisfies Laplace's equation, which implies that u ϵ has the mean value property. Taking the limit as ϵ → 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.
More generally, the same result holds for every distributional solution of Laplace's equation: If T ∈ D ′ ( Ω ) satisfies ⟨ T , Δ ϕ ⟩ = 0 for every ϕ ∈ C c ∞ ( Ω ) , then T = T u is a regular distribution associated with a smooth solution u ∈ C ∞ ( Ω ) of Laplace's equation.
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 Δ u = 0 , then the singular support of u is empty since the singular support of 0 is empty, meaning that u ∈ C ∞ ( Ω ) . In fact, since the Laplacian is elliptic, a stronger result is true, and solutions of Δ u = 0 are real-analytic.
