Behaviour of the first eigenvalue of the $p$-Laplacian in a domain with a hole
Tom 87 / 2001
Colloquium Mathematicum 87 (2001), 103-111
MSC: 35J65, 35P30.
DOI: 10.4064/cm87-1-6
Streszczenie
We investigate the behaviour of a sequence $\lambda _{s}$, $s=1,2,\dots, $ of eigenvalues of the Dirichlet problem for the $p$-Laplacian in the domains $% {\mit\Omega} _{s}$, $s=1,2,\dots, $ obtained by removing from a given domain ${\mit\Omega} $ a set $E_{s}$ whose diameter vanishes when $s\rightarrow \infty $. We estimate the deviation of $\lambda _{s}$ from the eigenvalue of the limit problem. For the derivation of our results we construct an appropriate asymptotic expansion for the sequence of solutions of the original eigenvalue problem.
