Whereas the optimal solvability of parabolic boundary value problems in strong settings is by now well understood, this is not so for the weak setting. It is well known that that the theory of weak solutions for elliptic and parabolic differential equations is of paramount importance for most boundary value problems of mathematical physics, for nonlinear reaction diffusion equations or the Navier Stokes equations of fluid dynamics, to name just two. However, up to now, there is no complete theory of maximal regularity for weak solutions of parabolic boundary value problems. This is true even in the â€simpleâ€ L_2 setting, in spite of the efforts of J-L. Lions and E. Magenes who developed an impressive machinery to solve this problem.

In recent work we have studied the problem of maximal regularity for parabolic evolution equations and not only complemented the Lions-Magenes theory but, more importantly, established the optimal weak solvability in the Lp-setting. Our approach is based on an in-depth study of the theory of anisotropic function spaces, in particular of the behavior of these spaces on manifolds with corners, and on the Marcinkiewicz multiplier theorem, among other things.

In this lecture series we shall explain how anisotropic function spaces occur quite naturally in this theory and present some of the main ideas and techniques of our approach.