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 $\partial\Omega\setminus \Gamma_-$toNeumanndataon$\partial\Omega\setminus \Gamma_+$. First we prove uniqueness results in three dimensions under some conditions such as $\overline{{\mathrm{\Gamma }}_{+}\cup {\mathrm{\Gamma }}_{-}}=\mathrm{\partial }\mathrm{\Omega }$. Next we survey uniqueness results in two dimensions for various elliptic systems for arbitrarily given ${\mathrm{\Gamma }}_{-}={\mathrm{\Gamma }}_{+}$. Our proof is based on complex geometric optics solutions which are constructed by a Carleman estimate.