ABSTRACT
This chapter discusses inverse conductivity problem with single measurement. For the infinite measurement problem there is well-established theory. For the finite measurement problem not much is known. However, there has been some progress in the theoretical and numerical parts of the problem. The chapter focuses on the recent progress in the theoretical part of the problem and also focuses on a numerical work. It reviews some well-known facts in the theory of layer potentials. The theory of layer potentials has been well-developed during the last few decades in relation with the boundary value problems for the harmonic equation on Lipschitz domains. H. Kang and J. K. Seo also obtained that any ball in three dimension is determined uniquely by one measurement. For these uniqueness, the Neumann data can be chosen arbitrarily. In deriving the results the representation formulae reviewed in the previous section play an essential role.
