Uniqueness for Inverse Boundary Value Problems by Dirichlet-to-Neumann Map on Subboundaries

We consider inverse boundary value problems for elliptic equations of second order of determining coefficients by Dirichlet-to-Neumann map on subboundaries, that is, the mapping from Dirichlet data supported on subboundary ${partial Omega setminus Gamma_{-}}$ to Neumann data on subboundary ${partial Omega setminus Gamma_{+}}$ . First we prove uniqueness results in three dimensions under some conditions such as ${overline{Gamma_{+}cupGamma_{-}}= partialOmega}$ Next we survey uniqueness results in two dimensions for various elliptic systems for arbitrarily given ${Gamma_{-} = Gamma_{+}}$ Our proof is based on complex geometric optics solutions which are constructed by a Carleman estimate.