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Domain identification for inverse problem via conformal mapping and fixed point methods in two dimensions

In this paper, we present a survey of the inverse eigenvalue problem for a Laplacian equation based on available Cauchy data on a known part $\Gamma_{0}$ and a homogeneous Dirichlet condition on an unknown part $\Gamma$ of the boundary of a bounded domain, $\Omega\subset{\mathbb R^N}$. We consider variations in the eigenvalues and propose a conformal mapping tool to reconstruct a part of the boundary curve of the two-dimensional bounded domain based on the Cauchy data of a holomorphic function that maps the unit disk onto the unknown domain. The boundary values of this holomorphic function are obtained by solving a non-local differential Bessel's equation. Then, the unknown boundary is obtained as the image of the boundary of the unit disk by solving an ill-posed Cauchy problem for holomorphic functions via a regularized power expansion. The Cauchy data were restricted to a non-vanishing function and to the normal derivatives without zeros. We prove the existence and uniqueness of the holomorphic function being considered and use the fixed-point method to numerically analyze the results of convergence. We'll calculate the eigenvalues and compare the result with the shape obtained via minimization functional method, as developed in a previous study. Further, we'll observe via simulations the shape of $\Gamma$, and if it preserves its properties with varying the eigenvalues.

Auteur(s) : Fagueye NDIAYE
Pages : 12
Année de publication : 2020
Revue : Hindawi "Abstract and Applied Analysis"
N° de volume : 2020
Type : Article
Statut Editorial : Open Access
Mise en ligne par : NDIAYE Faguèye