2020 Domain Identification for Inverse Problem via Conformal Mapping and Fixed Point Methods in Two Dimensions
Fagueye Ndiaye
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Abstr. Appl. Anal. 2020: 1-12 (2020). DOI: 10.1155/2020/1745656


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 Γ0 and a homogeneous Dirichlet condition on an unknown part Γ0 of the boundary of a bounded domain, ΩN . 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 nonlocal differential Bessel 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 nonvanishing 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 Γ and if it preserves its properties with varying the eigenvalues.


We would like to thank Editage (https://www.editage.com) for English language editing services.


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Fagueye Ndiaye. "Domain Identification for Inverse Problem via Conformal Mapping and Fixed Point Methods in Two Dimensions." Abstr. Appl. Anal. 2020 1 - 12, 2020. https://doi.org/10.1155/2020/1745656


Received: 25 April 2020; Accepted: 17 June 2020; Published: 2020
First available in Project Euclid: 28 July 2020

Digital Object Identifier: 10.1155/2020/1745656

Rights: Copyright © 2020 Hindawi


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