Abstract: This chapter aims to obtain basic estimates for the PME and related equations using as a main tool the techniques of symmetrization and mass concentration comparison, combined with scaling properties. Section 17.1 reviews the main concepts from symmetrization theory and introduces a comparison of mass concentrations. Sections 17.2 and 17.3 derive the basic comparison results for the elliptic equations to which the PME is reduced when it is solved by the ITD method. Comparison theorems for the evolution are then derived in Section 17.4. The application to the PME is in Section 17.5.

Keywords: PME, symmetrization, mass concentration, scaling,