Weyl's lemma (Laplace equation)

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

[math]\displaystyle{ \int_\Omega u(x) \, \Delta \varphi (x) \, dx = 0 }[/math]

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

This result implies the interior regularity of harmonic functions in [math]\displaystyle{ \Omega }[/math], but it does not say anything about their regularity on the boundary [math]\displaystyle{ \partial\Omega }[/math].

Idea of the proof

To prove Weyl's lemma, one convolves the function [math]\displaystyle{ u }[/math] with an appropriate mollifier [math]\displaystyle{ \varphi_\varepsilon }[/math] and shows that the mollification [math]\displaystyle{ u_\varepsilon = \varphi_\varepsilon\ast u }[/math] satisfies Laplace's equation, which implies that [math]\displaystyle{ u_\varepsilon }[/math] has the mean value property. Taking the limit as [math]\displaystyle{ \varepsilon\to 0 }[/math] and using the properties of mollifiers, one finds that [math]\displaystyle{ u }[/math] 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 [math]\displaystyle{ T\in D'(\Omega) }[/math] satisfies [math]\displaystyle{ \langle T, \Delta \varphi\rangle = 0 }[/math] for every [math]\displaystyle{ \varphi\in C_c^\infty(\Omega) }[/math], then [math]\displaystyle{ T= T_u }[/math] is a regular distribution associated with a smooth solution [math]\displaystyle{ u\in C^\infty(\Omega) }[/math] 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 [math]\displaystyle{ P }[/math] with smooth coefficients is hypoelliptic if the singular support of [math]\displaystyle{ P u }[/math] is equal to the singular support of [math]\displaystyle{ u }[/math] for every distribution [math]\displaystyle{ u }[/math]. The Laplace operator is hypoelliptic, so if [math]\displaystyle{ \Delta u = 0 }[/math], then the singular support of [math]\displaystyle{ u }[/math] is empty since the singular support of [math]\displaystyle{ 0 }[/math] is empty, meaning that [math]\displaystyle{ u\in C^\infty(\Omega) }[/math]. In fact, since the Laplacian is elliptic, a stronger result is true, and solutions of [math]\displaystyle{ \Delta u = 0 }[/math] are real-analytic.

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.