Non-autonomous Cauchy problems governed by forms: maximal regularity and invariance

Abstract

Form methods are a useful and elegant framework to study second order elliptic operators in divergence form. They can be used to describe such operators including various boundary conditions, such as Dirichlet, Neumann and Robin boundary conditions. In the autonomous case Cauchy problems of the form u´(t) + Au(t) = f (t), u(0) = u_0, where A is associated with a form a, are well studied. The subject of this thesis are non-autonomous Cauchy problems associated with a form a(t) depending on t. We study regularity, invariance of convex sets and asymptotics.