Harnack, Holder, Gauss and Widder: Serrin's Parabolic Legacy

TL;DR

This paper reviews and discusses J. Serrin's significant but less-known contributions to the theory of quasilinear parabolic equations, highlighting their impact on regularity, maximum principles, and fundamental solutions.

Contribution

It highlights Serrin's extensions of Moser's techniques to quasilinear parabolic equations and summarizes key results like maximum principles and Gaussian estimates.

Findings

Proved maximum principle and Hölder continuity for quasilinear parabolic equations.

Derived a Harnack principle for broad classes of equations.

Established Gaussian bounds for fundamental solutions.

Abstract

James Serrin's fundamental contributions to the theory of quasilinear elliptic equations are well-known and widely appreciated. He also made less well-known contributions to the theory of quasilinear parabolic equations which we dicuss in this note. Jurgen Moser gave greatly simplified proofs of the De Giorgi-Nash regularity results for linear divergence structure elliptic and parabolic differential equations using an original iterative technique. Serrin extended Moser's techniques and applied them to the study of divergence structure quasilinear elliptic equations, and in collaboration with Aronson, to divergence structure quasilinear parabolic equations. Specifically, among other results, they proved a maximum principle, Holder continuity of generalized solutions and derived a Harnack principle for a very broad class of quasilinear parabolic equations. In subsequent work, Aronson applied these results to study non-negative solutions to divergence structure linear equations without regularity assumptions on the coefficients. The results include a two-sided Gaussian estimate for the fundamental solution and a generalization of the Widder Representation Theorem.