A weighted symmetrization for nonlinear elliptic and parabolic equations in inhomogeneous media

Abstract

In the theory of elliptic equations, the technique of Schwarz symmetrization is one of the tools used to obtain a priori bounds for classical and weak solutions in terms of general information on the data. A basic result says that, in the absence of lower-order terms, the symmetric rearrangement of the solution $u$ of an elliptic equation, that we write $u^{\ast }$, can be compared pointwise with the solution of the symmetrized problem. The main question we address here is the modification of the method to take into account degenerate equations posed in inhomogeneous media. Moreover, the equations we want to deal with involve weights that make them of non-divergence form, at least when written in terms of the natural variables. We find comparison results covering the elliptic case and the corresponding evolution models of parabolic type, with emphasis on equations of porous medium type. More specifically, we obtain a priori bounds and decay estimates for wide classes of solutions of those equations.