Dr. Joachim Rehberg, WIAS Berlin

First, we motivate the notion 'maximal parabolic regularity’ as an instrument for the treatment for parabolic equations - linear and nonlinear ones. Then several fundamental properties of this are recalled. In the central part we give extremely general sufficient conditions on the geometry of the domain Ω, the Dirichlet boundary part D and the coefficient function μ for a second order divergence operator A to fulfill maximal parabolic regularity. After this we quote the famous theorem of Jan Prüss on quasilinear parabolic equations which provides local (in time) existence and uniqueness for such equations.
At the end we present several elliptic regularity results which then allow to treat real world problems as the semiconductor equations or the famous Keller-Segel model from Mathematical Biology.