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 $\partial\Omega\setminus \Gamma_-$ to Neumann data on $\partial\Omega\setminus \Gamma_+$. First we prove uniqueness results in three dimensions under some conditions such as $\bar{\Gamma_+ \cup \Gamma_-} = \partial\Omega$. 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.