A Note 0n the Equilibrium Potential of Certain Dirichlet Spaces

There exists an equilibrium potential, an element of a Dirichlet space, for any compact subset, with non-emtpy interior, of R ^m. This potential is constant on the interior of the set. There is a corresponding measure that is 0 outside the set. We prove that the restriction of this equilibrium measure to the interior of the set is absolutely continuous, and we derive an explicit formula for its density.