MATHEMATICA BOHEMICA, Vol. 139, No. 2, pp. 185-193, 2014

Spectrum of the Laplacian in narrow tubular neighbourhoods of hypersurfaces with combined Dirichlet and Neumann boundary conditions

David Krejčiřík

David Krejčiřík, Nuclear Physics Institute AS CR, Hlavní 130, 250 68 Řež, Czech Republic, e-mail:

Abstract: We consider the Laplacian in a domain squeezed between two parallel hypersurfaces in Euclidean spaces of any dimension, subject to Dirichlet boundary conditions on one of the hypersurfaces and Neumann boundary conditions on the other. We derive two-term asymptotics for eigenvalues in the limit when the distance between the hypersurfaces tends to zero. The asymptotics are uniform and local in the sense that the coefficients depend only on the extremal points where the ratio of the area of the Neumann boundary to the Dirichlet one is locally the biggest.

Keywords: Laplacian in tubes; Dirichlet boundary condition; Neumann boundary condition; eigenvalue asymptotics; dimension reduction; quantum waveguides; mean curvature

Classification (MSC 2010): 35P15, 49R05, 58J50, 81Q15

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