Entropy methods for parabolic and elliptic equations

Entropy methods associated with linear and nonlinear parabolic equations are powerful tools for establishing results of symmetry, unique- ness and optimality in functional inequalities. Optimality cases can be identified by considering asymptotic regimes and appropriate lineariza- tions. Entropy methods raise various issues of regularity and rely on integrations by parts which are not straightforward to justify. On the other hand, these methods give optimal criteria in some fundamental examples. This lecture will be devoted to an overview of the state of the art and list some open questions.