From the Polya-Szego symmetrization inequality for Dirichlet integrals to comparison theorems for p.d.e.'s on manifolds

A method for proving symmetrization inequalities for some elliptic p.d.e.'s on manifolds equipped with appropriate isoperimetric inequalities is outlined. The method is based on a modification of an approach of Baernstein. The question of what is the most general result that can be proved in this way is still open, and the author can be consulted if the reader is interested in this question.