The Dirichlet problem by variational methods

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.