Weyl's lemma (Laplace equation)

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In mathematics, Weyl's lemma is a result that provides a "very weak" form of the Laplace equation. It is named after the German mathematician Hermann Weyl.

1 Statement of the lemma
2 Proof
3 "Weak" and "very weak" forms of the Laplace equation
4 References

[edit] Statement of the lemma

Let $n \in \mathbb{N}$ and let $Ω$ be an open subset of $\mathbb{R}^{n}$ . Let $Δ$ denote the usual Laplace operator. Suppose that $u$ is locally integrable (i.e., $u \in L_{\mathrm{loc}}^{1} (\Omega; \mathbb{R})$ ) and that

$\int_{\Omega} u(x) \Delta \phi (x) \, \mathrm{d} x = 0$ $\quad$ (Eq. 1)

for every smooth function $\phi : \Omega \to \mathbb{R}$ with compact support in $Ω$ . Then, possibly after redefinition on a set of measure zero, $u$ is smooth and has $Δ u = 0$ in $Ω.$

[edit] Proof

Weyl's lemma follows from more general results concerning so-called the regularity property of elliptic operators. For example, one way to see why the lemma holds is to note that elliptic operators do not shrink singular support and that $0$ has no singular support.

[edit] "Weak" and "very weak" forms of the Laplace equation

The strong formulation of the Laplace equation is to seek functions $u$ with $Δ u = 0$ in some domain of interest, $Ω$ . The usual weak formulation is to seek weakly-differentiable functions $u$ such that

$\int_{\Omega} \nabla u (x) \cdot \nabla \phi (x) \, \mathrm{d} x = 0$ $\quad$ (Eq. 2)

for every $φ$ in the Sobolev space $W_{0}^{1, 2} (\Omega; \mathbb{R})$ . A solution of (Eq. 2) will also satisfy (Eq. 1) above, and the converse holds if, in addition, $u \in W^{1, 2} (\Omega; \mathbb{R})$ . Consequently, one can view (Eq. 1) as a "very weak" form of the Laplace equation, and a solution of (Eq. 1) as a "very weak" solution of $Δ u = 0$ .