Abstract
Let be an bounded open set and . Assume that has an extension such that . Then by
the Riesz representation theorem there exists a unique
We show that coincides with the Perron solution of the
Dirichlet problem
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.