Regularity in capacity and the Dirichlet Laplacian

摘要

Given an open set Omega in R-N, we prove that every function u in H-0(1)(Omega) boolean AND C((Omega) over bar) is zero everywhere on the boundary partial derivative Omega if and only if Omega is regular in capacity. If in addition Omega is bounded, then it is regular in capacity if and only if the mapping phi bar right arrow u(phi, Omega) from C(partial derivative Omega) into H(Omega) is injective, where u(phi, Omega) denotes the Perron solution of the Dirichlet problem. Let R be the set of all open subsets of R-N which are regular in capacity. Then one can define metrics d(l) and d(g) on R only involving the resolvent of the Dirichlet Laplacian. Convergence in those metrics will be defined to be the local/global uniform convergence of the resolvent of the Dirichlet Laplacian applied to the constant function 1. We prove that the spaces (R, d(g)) and ( R, d(l)) are complete and contain the set W of all open sets which are regular in the sense of Wiener ( or Dirichlet regular) as a closed subset.