Weyl's lemma (Laplace equation) explained

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

be an open subset of -dimensional Euclidean space , and let denote the usual Laplace operator. Weyl's lemma^[1] states that if a locally integrable function is a weak solution of Laplace's equation, in the sense that

for every test function (smooth function with compact support)

, then (up to redefinition on a set of measure zero) is smooth and satisfies pointwise in .

This result implies the interior regularity of harmonic functions in

, but it does not say anything about their regularity on the boundary .

Idea of the proof

To prove Weyl's lemma, one convolves the function

with an appropriate mollifier and shows that the mollification satisfies Laplace's equation, which implies that has the mean value property. Taking the limit as and using the properties of mollifiers, one finds that also has the mean value property,^[2] which implies that it is a smooth solution of Laplace's equation.^{[3] [4]} 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

satisfies for every , then is a regular distribution associated with a smooth solution of Laplace's equation.^[5]

Connection with hypoellipticity

Weyl's lemma follows from more general results concerning the regularity properties of elliptic or hypoelliptic operators.^[6] A linear partial differential operator

with smooth coefficients is hypoelliptic if the singular support of is equal to the singular support of for every distribution . The Laplace operator is hypoelliptic, so if , then the singular support of is empty since the singular support of is empty, meaning that . In fact, since the Laplacian is elliptic, a stronger result is true, and solutions of are real-analytic.

References

• Book: Gilbarg, David . Elliptic Partial Differential Equations of Second Order . Neil S. Trudinger . Neil Trudinger . 1988. Springer . 3-540-41160-7.
• Book: Stein, Elias . Elias Stein

. Elias Stein. 2005 . Real Analysis: Measure Theory, Integration, and Hilbert Spaces . Princeton University Press . 0-691-11386-6.

Notes and References

1. [Hermann Weyl]
2. The mean value property is known to characterize harmonic functions in the following sense. Let

   h\inC(\Omega)

   . Then

   h

   is harmonic in the usual sense (so

   h\inC^2(\Omega)

   and

   \left.\Deltah=0\right)

   if and only if for all balls

   B_{r\left(x0\right)}\Subset\Omega

   we have

   h\left(x
   	0\right)= 1 \omegan-1rn-1

   \int
   	\partialBr\left(x0\right)

   h(x)dS_x.

   where

   \omega_n-1r^n-1

   is the (n - 1)-dimensional area of the hypersphere

   \partialB_{r\left(x0\right)}

   .

   Using polar coordinates about

   x₀

   we see that when

   h

   is harmonic, then for

   B_{r\left(x0\right)}\Subset\Omega

   ,

   h\left(x_{0\right)=\left(\rhor}*h\right)\left(x_0\right).

3. Bernard Dacorogna, Introduction to the Calculus of Variations, 2nd ed., Imperial College Press (2009), p. 148.
4. Web site: Weyl’s lemma, one of many . Stroock . Daniel W..
5. [Lars Gårding]
6. [Lars Hörmander]