Regularity theory for free boundaries and transmission problems

Our purpose with this  project i twofold.

First to bridge the regularity theory of the  two directions of research,  free boundaries (FB) and transmission problems (TP), whenever appropriate. Second is to develop new tools to treat both problems in the case of system; hitherto almost untouched.

Both FB and TP have long history with great importance in  diverse areas in mathematics, where modelling of problems lead to partial differential equations with phase transition (FB) or transmission (TP).

The fundamentals of  both theories many times parallel or overlap and yet there are no unified approach to both problems whenever such overlap appears. Our second objective is to develop tools to treat both problems,  for the system case.

This project is devided into several sub-projects: FB include:  Global solutions and local singularities,  Problems with non-smooth r.h.s.,  Linear and Nonlinear  systems.

TP  include:  Eshelby´s conjecture,  Higher order regularity, TP with free boundaries.The time frame  is set for 4-years, and will be done in collaboration.These   projects  are in the mainstream of the theoretical FB and TP.

Through our study we expect the emergence of  new directions, and tools, as well as   answering many open questions of fundamental interest to both theoretical and applied community.

We believe the tools we create will have enough momentum to carry over into many problems related to systems, since currently tools for treating systems are underdeveloped.