SPECTRUM OF THE LAPLACIAN IN A NARROW CURVED STRIP WITH COMBINED DIRICHLET AND NEUMANN BOUNDARY CONDITIONS

0330433 - ÚJF 2010 RIV FR eng J - Journal Article
Krejčiřík, David
SPECTRUM OF THE LAPLACIAN IN A NARROW CURVED STRIP WITH COMBINED DIRICHLET AND NEUMANN BOUNDARY CONDITIONS.
ESAIM-Control Optimisation and Calculus of Variations. Roč. 15, č. 3 (2009), s. 555-568. ISSN 1292-8119. E-ISSN 1262-3377
R&D Projects: GA MŠMT LC06002
Institutional research plan: CEZ:AV0Z10480505
Keywords : Laplacian in tubes * Dirichlet and Neumann boundary conditions * dimension reduction
Subject RIV: BE - Theoretical Physics
Impact factor: 1.084, year: 2009

We consider the Laplacian in a domain squeezed between two parallel curves in the plane, subject to Dirichlet boundary conditions on one of the curves and Neumann boundary conditions on the other. We derive two-term asymptotics for eigenvalues in the limit when the distance between the curves 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 curvature radii of the Neumann boundary to the Dirichlet one is the biggest. We also show that the asymptotics can be obtained from a form of norm-resolvent convergence which takes into account the width-dependence of the domain of definition of the operators involved.
Permanent Link: http://hdl.handle.net/11104/0176222