Geodesic
Spectrum of the Laplacian in narrow tubular neighbourhoods of hypersurfaces with combined Dirichlet and Neumann boundary conditions
Mathematica Bohemica, Tome 139 (2014) no. 2, pp. 185-193

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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.
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.
DOI : 10.21136/MB.2014.143848
Classification : 35J05, 35P15, 49R05, 58J50, 81Q15
Keywords: Laplacian in tubes; Dirichlet boundary condition; Neumann boundary condition; eigenvalue asymptotics; dimension reduction; quantum waveguides; mean curvature 
Krejčiřík, David. Spectrum of the Laplacian in narrow tubular neighbourhoods of hypersurfaces with combined Dirichlet and Neumann boundary conditions. Mathematica Bohemica, Tome 139 (2014) no. 2, pp. 185-193. doi: 10.21136/MB.2014.143848
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