Calder\'on's problem for p-Laplace type equations

Abstract

We investigate a generalization of Calder\'on's problem of recovering the conductivity coefficient in a conductivity equation from boundary measurements. As a model equation we consider the p-conductivity equation with p strictly between one and infinity, which reduces to the standard conductivity equation when p equals two, and to the p-Laplace equation when the conductivity is constant. The thesis consists of results on the direct problem, boundary determination and detecting inclusions. We formulate the equation as a variational problem also when the conductivity may be zero or infinity in large sets. As a boundary determination result we recover the first order derivative of a smooth conductivity on the boundary. We use the enclosure method of Ikehata to recover the convex hull of an inclusion of finite conductivity and find an upper bound for the convex hull if the conductivity within an inclusion is zero or infinite.