A local regularization scheme of Cauchy problem for the Laplace equation on a doubly connected domain

The Cauchy problem of the Laplace equation is investigated for both exact and perturbed data on a doubly connected domain, i.e., the numerical reconstruction of the function value and the normal derivative value on a part of the boundary from the knowledge of exact or noisy Cauchy data on the remaining and accessible boundary, which is completely different from the Cauchy problem on a simply connected bounded region. We first establish the existence of a solution through the potential theory. By expressing the solution as a sum of single-layer potentials using boundary value condition, we get the integral equation systems about the density function on the boundary, and by applying local regularization scheme to the obtained integral equation systems, we get the regularization solution of the original problem. Some numerical results are presented to validate the applicability and effectiveness of the proposed method.