Proposition 12.4.14 (Green’s Identities).label Let $\OM\suf\R^{n}$ be a bounded domain with $C^{\infty}$ boundary $S\define \pd \OM$. Then for any $u,v\in C^{2}(\cl{\OM})$ we have
- (1)
$\integral{S }{}v\pd_{\nu} ud\s=\integral{\OM}{}(v\la u-\nabla v\cd \nabla u)dx$
- (2)
$\integral{S}{}(v\pd_{\nu} u-u\pd_{\nu} v)d\s=\integral{\OM}{}(v\la u-u\la v)dx$
Proof. This is a simple corollary of Gauss Divergence Theorem in $\R^{n}$ [1]
\begin{align*}\integral{S }{}(\vec{F}\vec{\nu})d\s=\integral{\OM}{}(\vec{\nabla}\cd \vec{F})dx\end{align*}
Take $\vec{F}=v\vec{\nabla}u$ then $\vec{\nabla}(v\vec{\nabla}u)=\vec{\nabla}v\cd \vec{\nabla}u-v\la u$$\square$
- everything is a corollary of Stoke’skeyboard_return
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