×

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
PDFBibTeX XMLCite
Full Text: DOI

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 · doi:10.1007/BF01206016
[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 · doi:10.1016/0022-1236(75)90028-2
[10] Simon, B.: Functional integration and quantum physics. New York: Academic Press 1979 · Zbl 0434.28013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.