Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4613562 | Journal of Differential Equations | 2006 | 24 Pages |
Abstract
We consider the first eigenvalue of the Dirichlet–Laplacian in three cases: C1,1-domains, Lipschitz domains, and bounded domains without any smoothness assumptions. Asymptotic formula for this eigenvalue is derived when domain subject arbitrary perturbations. For Lipschitz and arbitrary nonsmooth domains, the leading term in the asymptotic representation distinguishes from that in the Hardamard formula valid for smooth perturbations of smooth domains. For asymptotic analysis we propose and prove an abstract theorem demonstrating how eigenvalues vary under perturbations of both operator in Hilbert space and Hilbert space itself. This abstract theorem is of independent interest and has substantially broader field of applications.
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Physical Sciences and Engineering
Mathematics
Analysis