The Dirichlet problem by variational methods

Abstract

We consider the classical Dirichlet problem for harmonic functions on an open bounded set in ${ℝ}^N$ with continuous boundary data. Dirichlet’s principle provides a variational method to solve the problem. As an example by Hadamard shows, the Dirichlet principle is not applicable for all boundary data, but only for those which have an extension to a function $\Phi \in H^1\left(\Omega \right)$.

The aim of the talk is to show that there is a variational method to solve the Dirichlet problem even if there is no extension of the boundary data to $H^1\left(\Omega \right)$. We present an approach which works if there is an extension $\Phi$ such that $\Delta \Phi \in H^{-1}\left(\Omega \right)$. The method applies to Hadamard’s counter example and therefore provides a variational principle under much weaker assumptions than required for Dirichlet’s principle. This is joint work with W Arendt.