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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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[1] Bensoussan, A., Lions, J.L., Papanicolaou, G.C.: Asymptotic methods in periodic structures. Amsterdam: North-Holland 1978 · Zbl 0404.35001
[2] Huruslov, E.Ja., Marchenko, V.A.: Boundary value problems in regions with fine-grained boundaries (in Russian). Kiev 1974
[3] Kac, M.: Probabilistic methods in some problems of scattering theory. Rocky Mountain J. Math.4, 555-538 (1974) · Zbl 0314.47006
[4] Lions, J.L.: Some methods in mathematical analysis of systems and their control. New York: Gordon and Breach 1981 · Zbl 0542.93034
[5] Ozawa, S.: On an elaboration of M. Kac’s theorem concerning eigenvalues of the Laplacian in a region with random distributed small obstacles. Commun. Math. Phys.91, 473-487 (1983) · Zbl 0541.35019
[6] Ozawa, S.: Spectra of domains with small spherical Neumann boundary. J. Fac. Sci. Univ. Tokyo, Sect. IA30, 259-271 (1983) · Zbl 0541.35061
[7] Papanicolaou, G.C., Varadhan, S.R.S.: Diffusion in region with many small holes. In: Lecture Notes in Control and Information, Vol. 75, pp. 190-206. Berlin, Heidelberg, New York: Springer 1980 · Zbl 0485.60076
[8] Papanicolaou, G.C., Varadhan, S.R.S.: Boundary value problems with rapidly oscillating random coefficients. Coll. Math. Soc. János Bolyai 27, Random fields, Vol. II, pp. 835-873, Fritz, J., Lebowitz, J.L., Szász, D. (eds.). Amsterdam: North-Holland 1981 · Zbl 0499.60059
[9] Rauch, J., Taylor, M.: Potential and scattering theory on wildly perturbed domains. J. Funct. Anal.18, 27-59 (1975) · Zbl 0293.35056
[10] Simon, B.: Functional integration and quantum physics. New York: Academic Press 1979 · Zbl 0434.28013
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