Let Omega subset of R-n be a bounded domain. We perturb it to a domain Omega(epsilon) attaching a family of small protuberances with "room-and-passage"-like geometry (epsilon > 0 is a small parameter). Peculiar spectral properties of Neumann problems in so perturbed domains were observed for the first time by R. Courant and D. Hilbert. We study the case, when the number of protuberances tends to infinity as epsilon > 0 and they are E-periodically distributed along a part of partial derivative Omega. Our goal is to describe the behavior of the spectrum of the operator A(epsilon) = -(rho(epsilon))(-l) Delta(epsilon)(Omega), where Delta(epsilon)(Omega) is the Neumann Laplacian in Omega(epsilon), and the positive function rho(epsilon) is equal to 1 in Omega. We prove that the spectrum of A(epsilon) converges as epsilon -> 0 to the "spectrum" of a certain boundary value problem for the Neumann Laplacian in Omega with boundary conditions containing the spectral parameter in a nonlinear manner. Its eigenvalues may accumulate to a finite point. (C) 2015 Elsevier Inc. All rights reserved. ZB 0 Z8 0 ZR 0 ZS 0

Neumann spectral problem in a domain with very corrugated boundary

Cardone G;
2015

Abstract

Let Omega subset of R-n be a bounded domain. We perturb it to a domain Omega(epsilon) attaching a family of small protuberances with "room-and-passage"-like geometry (epsilon > 0 is a small parameter). Peculiar spectral properties of Neumann problems in so perturbed domains were observed for the first time by R. Courant and D. Hilbert. We study the case, when the number of protuberances tends to infinity as epsilon > 0 and they are E-periodically distributed along a part of partial derivative Omega. Our goal is to describe the behavior of the spectrum of the operator A(epsilon) = -(rho(epsilon))(-l) Delta(epsilon)(Omega), where Delta(epsilon)(Omega) is the Neumann Laplacian in Omega(epsilon), and the positive function rho(epsilon) is equal to 1 in Omega. We prove that the spectrum of A(epsilon) converges as epsilon -> 0 to the "spectrum" of a certain boundary value problem for the Neumann Laplacian in Omega with boundary conditions containing the spectral parameter in a nonlinear manner. Its eigenvalues may accumulate to a finite point. (C) 2015 Elsevier Inc. All rights reserved. ZB 0 Z8 0 ZR 0 ZS 0
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.12070/5858
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