How is it the Dirichlet's principle on compact manifolds with boundary $\partial M= \mbox{manifold boundary}$?. I've just found the Dirichlet's principle on domains $U \subset \mathbb{R}^{n}$, i.e, $U$ is open, bounded, and $\partial U$ is $C^{1}$, where $\partial U= \mbox{is the topology boundary}$, and $u \in C^{2}(\bar U)$ solves $-\Delta u=f \quad \mbox{in} \quad U $, and $u=g \quad \mbox{in}\quad \partial U$, iff, $$I[u]=\mbox{min}_{w \in A} I[w]$$, where $I[w]:= \int_{U}\frac{1}{2}|\nabla w|^{2}- wf dx $ and $A:=\{w \in C^2(\bar{U}) : w=g \in \partial U \}.$

Is there anyone who can give me a reference about it?


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