Abstract:In this paper, we consider the asymptotic behavior for the principal eigenvalue of an elliptic operator with piecewise constant coefficients. This problem was first studied by Friedman in 1980. We show how the geometric shape of the interface affects the asymptotic behavior for the principal eigenvalue. This is a refinement of the result by Friedman.
“…This inverse problem is closely related to the coating problem and reinforcement problem [6,14,7,21,23]. If we consider a two-phase eigenvalue problem with a thin coating of the boundary with Dirichlet boundary condition, then Friedman [14] proved that the principal eigenvalue converges to the principal eigenvalue of a Robin eigenvalue problem.…”
Section: Introductionmentioning
confidence: 99%
“…The inverse problem 1.1 is closely related to the coating problem and reinforcement problem [2][3][4][5][6][7]. Let D ⊂ R n (n 2) be a bounded domain with smooth and connected boundary Γ.…”
We consider the problem of the recovery of a Robin coefficient on a part γ ⊂ ∂Ω of the boundary of a bounded domain Ω from the principal eigenvalue and the boundary values of the normal derivative of the principal eigenfunction of the Laplace operator with Dirichlet boundary condition on ∂Ω \ γ. We prove uniqueness, as well as local Lipschitz stability of the inverse problem. Moreover, we present an iterative reconstruction algorithm with numerical computations in two dimensions showing the accuracy of the method.
“…This inverse problem is closely related to the coating problem and reinforcement problem [6,14,7,21,23]. If we consider a two-phase eigenvalue problem with a thin coating of the boundary with Dirichlet boundary condition, then Friedman [14] proved that the principal eigenvalue converges to the principal eigenvalue of a Robin eigenvalue problem.…”
Section: Introductionmentioning
confidence: 99%
“…The inverse problem 1.1 is closely related to the coating problem and reinforcement problem [2][3][4][5][6][7]. Let D ⊂ R n (n 2) be a bounded domain with smooth and connected boundary Γ.…”
We consider the problem of the recovery of a Robin coefficient on a part γ ⊂ ∂Ω of the boundary of a bounded domain Ω from the principal eigenvalue and the boundary values of the normal derivative of the principal eigenfunction of the Laplace operator with Dirichlet boundary condition on ∂Ω \ γ. We prove uniqueness, as well as local Lipschitz stability of the inverse problem. Moreover, we present an iterative reconstruction algorithm with numerical computations in two dimensions showing the accuracy of the method.
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