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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 $$\Gamma$$.

MSC:
 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
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References:
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