Ricerc@Sapienza

The project focuses on a selection of nonlinear Partial Differential Equations (PDEs) which have at the same time a novel and challenging structure and a robust connection to models in Science and Engineering:

A - the 1-harmonic flow, a well-grounded tool in Image Processing which is also interesting per se, as a prototype of manifold-constrained gradient flows in BV spaces;

B - variants of the total variation flow, such as the (nonlinear) relativistic heat equation, which find applications whenever a-priori bounds on speed or flux of a diffusion process are modeling-wise relevant, thus sharing interesting qualitative features with nonlinear conservation laws;

C - fractional laplacian equations, an extremely active field in current mathematical research which arises in the modeling of nonlocal interactions, such as for the fractional Schroedinger equation;

D - free boundary problems for the thin-film equations, a fourth-order degenerate parabolic equation modelling wetting phenomena at "small" scales.

Our ambition is to give more insight on specific issues of relevance to applications while at the same time innovating mathematics, possibly achieving advances in the general theory of PDEs. Depending on the problem under consideration, our focus will be on well-posedness, regularity, and/or qualitative properties of solutions. In most cases, however, these three issues are strictly related to each other.

Our approach is strongly collaborative and inclined towards training. The project's goals will be pursued together with both young, talented researchers and more senior, internationally visible scientists. In addition, most topics are appropriate for post-doc research programs. The budget is distributed accordingly: besides overhead and a basic support to the Team members' mobility, funds are only requested for co-working with external collaborators and co-financing post-doc positions (and related equipment).