Comparison results for elliptic and parabolic equations via symmetrization: a new approach

We present here a new approach for comparison results of solutions of secondorder elliptic or parabolic equations by Schwarz symmetrization.This approach relies only on the fact that the fundamental solution of the heat equation in the whole space is spherically symmetric and decreasing and the proofs use, in addition to this well-known fact, a classical domination relationship and Trotter-Kato formula.We also apply this method to other symmetrizations like, for instance, Steiner symmetrization for which we thus derive some new comparison results. I. Introduction.It is by now well known that sharp bounds for solutions of elliptic and parabolic equations may be obtained using Schwarz symmetrization (i.e., the spherical nonincreasing rearrangement).Indeed, for large classes of equations, the solution may be "compared" to the solution of an analogous problem with spherical symmetry (the so-called symmetrized problems).The first results in this direction were obtained by H. Weinberger [40], G. Talenti [36], C. Bandle [8] and since then, have been extended in various directions by various authors: let us mention for instance A. Alvino and G. Trombetti [5]; P.L. Lions [24]; G. Chiti [13]; J.L. Vasquez [39], C. Bandle [9]; J. Mossino and J.M