Looking for best constants in inequalities between Hessians

In my talk I will introduce a family of functional inequalities introduced recently by myself and collaborators. It turned out to be fruitful in dealing with thermodynamics related systems of PDEs. I will emphasize the meaning of the magnitude of the best constant in such inequalities in context of new estimates in PDEs as well as information theory.

 

Finally, I will show two proofs of the family of such inequalities, the second yielding a better (but not yet sufficient:>) constant.