Approximate multiplicative controllability for degenerate parabolic problems and regularity properties of elliptic and parabolic systems

This thesis consists of two parts, both related to the theory of parabolic equations and systems. The first part is devoted to control theory which studies the possibility of influencing the evolution of a given system by an external action called control. Here we address approximate controllability problems via multiplicative controls, motivated by our interest in some differential models for the study of climatology. In the second part of the thesis we address regularity issues on the local differentiability and H\"older regularity for weak solutions of nonlinear systems in divergence form. In order to improve readability, the two parts have been organized as completely independent chapters, with two separate introductions and bibliographies. All the new results of this thesis have been presented at conferences and workshops, and most of them appeared or are to appear as research articles in international journals. Related directions for future research are also outlined in body of the work.