A convergence theorem for Dirichlet forms with applications to boundary value problems with varying domains (Q1895787)

It is shown that resolvents of Dirichlet boundary problems depend on the domain in a continuous manner. More precisely, two notions of convergence of the associated operators are investigated: Convergence in the strong resolvent sense is proven using monotone convergence of forms and a suitable representation of Dirichlet boundary conditions. To establish norm resolvent convergence we assume that the domains vary only in a region of finite capacity. Basic to the continuity with respect to norm resolvent sense is a convergence theorem for measure perturbations of Dirichlet forms, which is the main new result of the present article.