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A convergence theorem for Dirichlet forms with applications to boundary value problems with varying domains. (English) Zbl 0822.31005
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.

MSC:
31C25 Dirichlet forms
60J45 Probabilistic potential theory
47A40 Scattering theory of linear operators
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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