## 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.