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Zbl 1167.35011
Arendt, Wolfgang; Daners, Daniel
The Dirichlet problem by variational methods. (English)
[J] Bull. Lond. Math. Soc. 40, No. 1, 51-56 (2008). ISSN 0024-6093; ISSN 1469-2120

Summary: Let $\Omega\subset\Bbb R^N$ be a bounded open set and $\varphi\in C(\partial\Omega)$. Assume that $\varphi$ has an extension $\Phi\in C(\overline\Omega)$ such that $\Delta\Phi\in H^{-1}(\Omega)$. Then by the Riesz representation theorem there exists a unique $$u\in H_0^1(\Omega) \quad\text{such that}\quad -\Delta u=\Delta\Phi\quad\text{in }{\cal D}(\Omega)'.$$ We show that $u+\Phi$ coincides with the Perron solution of the Dirichlet problem $$\Delta h=0, \qquad h|_{\partial\Omega}=\varphi.$$ This extends recent results by {\it S. Hildebrandt} [Math. Nachr. 278, No. 1--2, 141--144 (2005; Zbl 1157.35340)] and {\it C. G.. Simader} [Math. Nachr. 279, No.~4, 415--430 (2006; Zbl 1163.35003)], and also gives a possible answer to Hadamard's objection against Dirichlet's principle.

MSC 2000:: *35J25 Second order elliptic equations, boundary value problems
35J05 Laplace equation, etc.
35J20 Second order elliptic equations, variational methods
31B20 Boundary value and inverse problems (higher-dim. potential theory)

Keywords: Dirichlet problem; Perron solution; Hadamard's objection; variational methods

Citations: Zbl 1157.35340; Zbl 1163.35003

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