Rigidity for Pólya-Szegö and perimeter inequalities under symmetrisation

The scope of the present thesis is the study of rigidity for Pólya-Szegö and perimeter inequalities under symmetrisation. The term rigidity refers to the situation in which all the extremals of the aforementioned inequalities are symmetric.

In the first part, we establish a Pólya-Szegö inequality for circular symmetrisation in any dimension and under general assumptions, both on the integrand and on the class of functions to which this can be applied. Moreover, we provide sufficient conditions for rigidity.

In the second part, we consider the perimeter inequality under Schwarz symmetrisation. In this case, we investigate the properties of the corresponding distribution function and we exploit several counterexamples, where rigidity is violated. Finally, we are able to find necessary and sufficient conditions for rigidity.