Weyl's lemma (Laplace equation)

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.

tatement of the lemma

Let $n\; in\; mathbb\{N\}$ and let $Omega$ be an open subset of $mathbb\{R\}^\{n\}$. Let $Delta$ 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 $phi\; :\; Omega\; o\; mathbb\{R\}$ with compact support in $Omega$. Then, possibly after redefinition on a set of measure zero, $u$ is smooth and has $Delta\; u\; =\; 0$ in $Omega.$

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.

"Weak" and "very weak" forms of the Laplace equation

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

:$int\_\{Omega\}\; abla\; u\; (x)\; cdot\; abla\; phi\; (x)\; ,\; mathrm\{d\}\; x\; =\; 0$ $quad$ (Eq. 2)

for every $phi$ 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 $Delta\; u\; =\; 0$.

References

*