The encloure method for semilinear elliptic equations with power type nonlinearities

Abstract

We employ the enclosure method to reconstruct unknown inclusions within an object that is governed by a semilinear elliptic equation with power-type nonlinearity. Motivated by [16], we tried to solve the problem without using special solutions such as complex geometrical optics solutions to the equation. Instead, we construct approximate solutions obtained from Taylor approximation of the solution operator. By incorporating such approximate solutions into the definition of an indicator functional, we are able to use the classical Calder\'{o}n-type harmonic functions to accomplish the reconstruction task.