Domain identification for harmonic functions

Abstract

The authors investigate the problem of identifying the domain G of a harmonic function u such that Cauchy data are given on a known portion of the boundary of G, while a zero Dirichlet condition is specified on the remaining portion of the boundary, which is to be found. Under certain conditions on the domain G, it is shown that the problem reduces to identifying the coefficients of an elliptic equation which, in turn, is converted into the problem of minimizing a functional. Under certain conditions on G, it is shown that the solution, if it exists, is unique. An application is pointed out for the problem of designing a vessel shape that realizes a given plasma shape. © 1995 Kluwer Academic Publishers.