Abstract
For any decreasing sequence of bounded finely open sets Di ⊂ RN it is shown that, for every n, the nth eigenvalue λn ( Di) of the Dirichlet laplacian A ( Di ) on Di converges to λn ( D ) (the nth eigenvalue of A ( D ) ), where D denotes the fine interior of ∩ Di. Likewise, A ( Di )-1 → A ( D )-1 in operator norm. Similar results are obtained for increasing or just order convergent sequences ( Di ). Furthermore, A ( D )-1 is identified with the integral operator on L2 ( D ) whose kernel is Green's function for D.
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Fuglede, B. The Dirichlet Laplacian on Finely Open Sets. Potential Analysis 10, 91–101 (1999). https://doi.org/10.1023/A:1008630909423
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DOI: https://doi.org/10.1023/A:1008630909423