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Random media and eigenvalues of the Laplacian. (English) Zbl 0555.35101
We consider a bounded domain \(\Omega\) in \(R^ 3\) with smooth boundary \(\Gamma\). We put \(B(\epsilon;w)=\{x\in R^ 3;\quad | x-w| <\epsilon \}.\) Fix \(\beta\geq 1\). Let \(0<\mu_ 1(\epsilon;w(m))\leq \mu_ 2(\epsilon;w(m))\leq...\) be the eigenvalues of -\(\Delta\) in \(\Omega_{\epsilon,w(m)}=\Omega \setminus \cup^{\tilde m}_{i=1}B(\epsilon;w_ i^{(m)})\) under the Dirichlet condition on its boundary. Here \(\tilde m\) denotes the largest integer which does not exceed \(m^{\beta}\), and w(m) denotes the set of m-points \(\{w_ i^{(m)}\}^{\tilde m}_{i=1}\in \Omega^{\tilde m}.\)
Let \(V(x)>0\) be a \(C^ 1\)-class function on \({\bar \Omega}\) satisfying \(\int_{\Omega}V(x)dx=1\). We consider \(\Omega\) as the probability space with the probability density V(x)dx. Let \(\Omega^{\tilde m}=\prod^{\tilde m}_{i=1}\Omega\) be the probability space with the product measure.
Theorem: Assume that \(1\leq \beta <9/8\) and \(V(x)>0\). Fix \(\alpha >0\) and k. Then, there exists a constant \(\delta (\beta)>0\) independent of m such that \[ \lim_{m\to \infty}{\mathbb{P}}(w(m)\in \Omega^{\tilde m};\quad m^{\delta '-(\beta -1)}| \mu_ k(\alpha /m;w(m))-\mu^ V_{k,m}| <\epsilon)=1 \] holds for any \(\epsilon >0\) and \(\delta '\in [0,\delta (\beta)).\) Here \(\mu^ V_{k,m}\) denotes the k-th eigenvalue of \(-\Delta +4\pi \alpha m^{\beta -1}V(x)\) in \(\Omega\) under the Dirichlet condition on \(\Gamma\).

35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35R60 PDEs with randomness, stochastic partial differential equations
Full Text: DOI
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