The Dirichlet Laplacian on finely open sets (Q1286333)

Let \(D\) be an open bounded set in \({\mathbb R}^N\), \(A(D)\) be the Dirichlet Laplacian in \(L^2(D)\). Denote by \(\lambda_n(D)\) the \(n\)th eigenvalue of \(A(D)\). Suppose that \(D\) is the interior of \(\cap_iD_i\), where \((D_i)_i\) is a decreasing sequence of bounded sets which are open in the Cartan fine topology (the weakest topology in which all subharmonic functions are continuous). The author shows that \(\lambda_n(D_i)\rightarrow \lambda_n(D)\) as \(i\rightarrow\infty\), and \(A(D_i)^{-1}\rightarrow A(D)^{-1}\) in operator norm. Similar results are obtained for increasing or just order convergent sequences \((D_i)_i\).