Carleman estimates for parabolic equations and applications

In this review, concerning parabolic equations, we give self-contained descriptions on (1) derivations of Carleman estimates; (2) methods for applications of the Carleman estimates to estimates of solutions and to inverse problems. Moreover limited to parabolic equations, we survey the previous and recent results in view of the applicability of the Carleman estimate. We do not intend to pursue any general treatments of the Carleman estimate itself but by showing it in a direct manner, we mainly aim to demonstrate the applicability of the Carleman estimate to the estimation of solutions and inverse problems.