Direct and inverse problem for geometric perturbation of the Laplace operator in a strip

This paper is concerned with the inverse problem of determining geometric shape of a part γ of the boundary of a perturbed strip Ω from a pair of Cauchy data of a harmonic function u in Ω. This leads to the study of the direct problem. Using the variational method, we show that is well posed, and by the integral equation method we seek the solution in the form of combined double- and single-layer potential. For the identification of γ we prove a uniqueness result, that is, a pair of Cauchy data on the accessible part \(\varGamma _{0}\) uniquely determines the missing part γ of the boundary, and we derive a system of nonlinear integral equations equivalent to our inverse problem. We present numerical examples for both the direct and inverse problems.