Degenerate Elliptic and Parabolic Equations and its Applications to Front Propagation*

Start Date: 2011 End Date: 2014
PIs: Diogo Gomes (IST/UTL), Dejan Slepcev (CMU)
Team: Instituto Superior Técnico of the Universidade Técnica de Lisboa (IST/UTL), Carnegie Mellon University (CMU)

This project builds upon collaborations between Portuguese researchers and CMU and UTAustin researchers in the framework of the CMU-Portugal and UTAustin-Portugal programs. It has two primary goals; the study of the partial differential equations that arise in front propagation, namely degenerate elliptic and parabolic equations, as well as related problems. The second is the applica- tions of these methods to more concrete problems, including, for instance geophysical applications (such as the modeling of ocean fronts), and the development of tools for the numerical analysis of inverse problems in front propagation.
We foresee that many of the techniques developed in this project will be of interest for other problems, such as mathematical finance (utility based valuation methods), non-linear filtering, classical mechanics (Aubry-Mather theory and its extensions), mathematical biology, and mean field games, homogenization and stochastic partial differential equations. In addition we foresee the study of coarsening systems, both in their own right and as models for more complex natural coarsening systems.
Several ICTI CMU-Portugal postdocs have been or are presently collaborating in this project, Filippo Cagnetti and Mohammad El Smaily, at IST with Diogo Gomes, and L. Monsaingeon and X.Lu at CMU.
This report covers the activities developed in collaboration with CMU faculty and only partially the activities in collaboration with UT Austin.

*Project approved in the Call 2009 in Applications of Mathematics Thematic Areas, in the framework of the Carnegie Mellon Portugal Program and UT Austin-Portugal