Domain Identification for Inverse Problem via Conformal Mapping and Fixed Point Methods in Two Dimensions

Abstract

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 ${\Gamma }_0$ of the boundary of a bounded domain, $\Omega \subset {ℝ}^N$ . 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 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 $\Gamma$ and if it preserves its properties with varying the eigenvalues.