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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
##### Keywords:
random media; eigenvalues; Dirichlet condition
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##### References:
 [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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