Weyl's lemma (Laplace 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

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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

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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.

Proof

Let   be the standard mollifier.

Fix a compact set   and put   be the distance between   and the boundary of  .

For each   and   the function

 

belongs to test functions   and so we may consider

 

We assert that it is independent of  . To prove it we calculate   for  .

Recall that

 

where the standard mollifier kernel   on   was defined at Mollifier#Concrete_example. If we put

 

then  .

Clearly   satisfies   for  . Now calculate

 

Put   so that

 

In terms of   we get

 

and if we set

 

then   with   for  , and  . Consequently

 

and so  , where  . Observe that  , and

 

Here   is supported in  , and so by assumption

 .

Now by considering difference quotients we see that

 .

Indeed, for   we have

 

in   with respect to  , provided   and   (since we may differentiate both sides with respect to  . But then  , and so   for all  , where  . Now let  . Then, by the usual trick when convolving distributions with test functions,

 

and so for   we have

 .

Hence, as   in   as  , we get

 .

Consequently  , and since   was arbitrary, we are done.

Generalization to distributions

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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

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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.

Notes

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  1. ^ Hermann Weyl, The method of orthogonal projections in potential theory, Duke Math. J., 7, 411–444 (1940). See Lemma 2, p. 415
  2. ^ The mean value property is known to characterize harmonic functions in the following sense. Let  . Then   is harmonic in the usual sense (so   and   if and only if for all balls   we have
     
    where   is the (n − 1)-dimensional area of the hypersphere  . Using polar coordinates about   we see that when   is harmonic, then for  ,
     
  3. ^ Bernard Dacorogna, Introduction to the Calculus of Variations, 2nd ed., Imperial College Press (2009), p. 148.
  4. ^ Stroock, Daniel W. "Weyl's lemma, one of many" (PDF).
  5. ^ Lars Gårding, Some Points of Analysis and their History, AMS (1997), p. 66.
  6. ^ Lars Hörmander, The Analysis of Linear Partial Differential Operators I, 2nd ed., Springer-Verlag (1990), p.110

References

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