Georgia Institute of Technology College of Sciences

In this talk I will present recent work, in collaboration with J.Dolbeault, G. Tarantello and A. Tertikas,about the symmetry properties of extremal functions for (interpolation)functional inequalities playing an important rolein the study of long time behavior of evolution diffusion equations.Optimal constants are rarely known,in fact one can write them explicitely only when the extremals enjoymaximal symmetry. This is why the knowledge of the parameters' regionswhere symmetry is achieved is of big importance. In the case of symmetrybreaking, the underlying phenomena permitting it are analyzed.

The McK--V system is a non--linear diffusion equation with a non--local non--linearity provided by convolution. Recently popular in a variety of applications, it enjoys an ancient heritage as a basis for understanding equilibrium and near equilibrium fluids. The model is discussed in finite volume where, on the basis of the physical considerations, the correct scaling (for the model itself) is identified. For dimension two and above and in large volume, the phase structure of the model is completely elucidated in (somewhat disturbing) contrast to dynamical results. This seminar represents joint work with V. Panferov.