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Eigenvalue problems on domains with cracks. I. (English) Zbl 1134.35319
Summary: We study an eigenvalue problem for the Laplace operator on a planar region with a growing crack. We impose Neumann boundary conditions on the crack and Dirichlet boundary conditions elsewhere. One tip of the crack is fixed at the boundary. We obtain full asymptotic expansions of the first two eigenvalues of the Laplace operator as the other tip of the crack reaches the boundary.
MSC:
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35C20 Asymptotic expansions of solutions to PDEs
35P15 Estimates of eigenvalues in context of PDEs
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