Regularity results for asymptotic problems

Elliptic and parabolic equations arise in the mathematical description of a wide variety of phenomena, not only in the natural science but also in engineering and economics. To mention few examples, consider problems arising in different contexts: gas dynamics, biological models, the pricing of assets in economics, composite media. The importance of these equations from the applications' point of view is equally interesting from that of analysis, since it requires the design of novel techniques to attack the always valid question of existence, uniqueness and regularity of solutions. \ In particular, in recent years parabolic problems came more and more into the focus of mathematicians. Changing from elliptic to the parabolic case means physically to switch from the stationary to the non-stationary case, i.e. the time is introduced as an additional variable. Exactly this natural origin constitutes our interest in parabolic problems: they reflect our perception of space and time. Therefore they often can be used to model physical process, e.g. heat conduction or diffusion process. medskip oindent In this thesis I will principally concentrate on the regularity properties of solutions of second order systems of partial differential equations in the elliptic and parabolic context. The outline of the thesis is as follows. smallskip oindent After giving some preliminary results, in the 3st Chapter we consider the parabolic analogue of some regularity results already known in the elliptic setting, concerning systems becoming parabolic only in an {it asymptotic} sense. In the standard elliptic version, these results prove the {it Lipschitz regularity} of solutions to elliptic systems of the type dive a(Du)=0, with u: Omega ightarrow R^{N}, under the main assumption that the vector field