The Dirichlet problem by variational methods

Preprint February 2007
Bulletin of the London Mathematical Society, 40 (2008), 51 - 56.
doi:10.1112/blms/bdm091

Abstract

Let $$\Omega \subset \mathbb R^N$$ be an bounded open set and $$\varphi \in C(\partial \Omega)$$. Assume that $$\varphi$$ has an extension $$\Phi \in C(\bar{\Omega})$$ such that $$\Delta \Phi \in H^{-1}(\Omega)$$. Then by the Riesz representation theorem there exists a unique $u \in H^1_0(\Omega) \quad\text{such that} \quad-\Delta u=\Delta\Phi \quad\text{in $$\mathcal D(\Omega)^\prime$$.}$ We show that $$u+\Phi$$ coincides with the Perron solution of the Dirichlet problem $\Delta h=0,\quad h|_{\partial \Omega}= \varphi.$ This extends recent results by Hildebrandt [Math. Nachr. 278 (2005), 141--144] and Simader [Math. Nachr. 279 (2006), 415--430], and also gives a possible answer to Hadamard's objection against Dirichlet's principle.

AMS Subject Classification (2000): 35J05, 31B05

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