A new proof of Moser's parabolic harnack inequality using the old ideas of Nash

Abstract

In 1958 Nash published his fundamental work on the local Holder continuity of solutions of second-order parabolic equations with coefficients which need not be smooth ([8]). The primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution. Though the work is often cited in the literature about weak solutions of elliptic and parabolic equations, one feels that Nash’S ideas were never fully understood (and maybe still are not) and that because of this the more understandable and seemingly more fruitful ideas of De Giorgi ([4]) and Moser ([6], [7]) were subsequently adopted.