View source for Weyl's lemma (Laplace equation)

{{Short description|Mathematical equation}}
In [[Mathematics|mathematics]], '''Weyl's lemma''', named after [[Biography:Hermann Weyl|Hermann Weyl]], states that every [[Weak solution|weak solution]] of [[Laplace's equation]] is a smooth solution. This contrasts with the [[Physics:Wave equation|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>\Omega</math> be an [[Open set|open subset]] of <math>n</math>-dimensional Euclidean space <math>\mathbb{R}^{n}</math>, and let <math>\Delta</math> denote the usual [[Laplace operator]]. Weyl's lemma<ref>[[Biography:Hermann Weyl|Hermann Weyl]], The method of orthogonal projections in potential theory, ''Duke Math. J.'', 7, 411–444 (1940). See Lemma 2, p. 415</ref> states that if a locally integrable function <math>u \in L_{\mathrm{loc}}^{1}(\Omega)</math> is a weak solution of Laplace's equation, in the sense that

:<math>\int_\Omega u(x) \, \Delta \varphi (x) \, dx = 0</math> 

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

This result implies the interior regularity of harmonic functions in <math>\Omega</math>, but it does not say anything about their regularity on the boundary <math>\partial\Omega</math>. 

== Idea of the proof ==
To prove Weyl's lemma, one [[Convolution|convolves]] the function <math>u</math> with an appropriate [[Mollifier|mollifier]] <math>\varphi_\varepsilon</math> and shows that the mollification <math>u_\varepsilon = \varphi_\varepsilon\ast u</math> satisfies Laplace's equation, which implies that <math>u_\varepsilon</math> has the mean value property. Taking the limit as <math>\varepsilon\to 0</math> and using the properties of mollifiers, one finds that <math>u</math> also has the mean value property, which implies that it is a smooth solution of Laplace's equation.<ref>Bernard Dacorogna, ''Introduction to the Calculus of Variations,'' 2nd ed., Imperial College Press (2009), p. 148.</ref> 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 [[Distribution (mathematics)|distributional solution]] of Laplace's equation: If <math>T\in D'(\Omega)</math> satisfies <math>\langle T, \Delta \varphi\rangle = 0</math> for every <math>\varphi\in C_c^\infty(\Omega)</math>, then <math>T= T_u</math> is a regular distribution associated with a smooth solution <math>u\in C^\infty(\Omega)</math> of Laplace's equation.<ref>Lars Gårding, ''Some Points of Analysis and their History'', AMS (1997), p. 66.</ref>

== Connection with hypoellipticity ==
Weyl's lemma follows from more general results concerning the regularity properties of elliptic or hypoelliptic operators.<ref>[[Biography:Lars Hörmander|Lars Hörmander]], ''The Analysis of Linear Partial Differential Operators I'', 2nd ed., Springer-Verlag (1990), p.110</ref> A linear partial differential operator <math>P</math> with smooth coefficients is hypoelliptic if the [[Support (mathematics)|singular support]] of <math>P u</math> is equal to the singular support of <math>u</math> for every distribution <math>u</math>. The Laplace operator is hypoelliptic, so if <math>\Delta u = 0</math>, then the singular support of <math>u</math> is empty since the singular support of <math>0</math> is empty, meaning that <math>u\in C^\infty(\Omega)</math>. In fact, since the Laplacian is elliptic, a stronger result is true, and solutions of  <math>\Delta u = 0</math> are [[Analytic function|real-analytic]].

==Notes==
{{reflist}}

==References==
*{{cite book |title= Elliptic Partial Differential Equations of Second Order |last=Gilbarg |first=David |author2=Neil S. Trudinger | year=1988| publisher=Springer |isbn=3-540-41160-7}}
*{{cite book | first=Elias | last=Stein |  year=2005 | title=Real Analysis: Measure Theory, Integration, and Hilbert Spaces | publisher=Princeton University Press | isbn=0-691-11386-6}}

[[Category:Partial differential equations]]
[[Category:Harmonic functions]]

{{Sourceattribution|Weyl's lemma (Laplace equation)}}