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.
be an open subset of
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)
This result implies the interior regularity of harmonic functions in
To prove Weyl's lemma, one convolves the function
and shows that the mollification
satisfies Laplace's equation, which implies that
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.
Fix a compact set
Recall that where the standard mollifier kernel
, and so by assumption Now by considering difference quotients we see that Indeed, for
(since we may differentiate both sides with respect to
Then, by the usual trick when convolving distributions with test functions, and so for
More generally, the same result holds for every distributional solution of Laplace's equation: If
is a regular distribution associated with a smooth solution
[5] 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
The Laplace operator is hypoelliptic, so if
is empty since the singular support of
In fact, since the Laplacian is elliptic, a stronger result is true, and solutions of