Heat kernel estimates for fourth order non-uniformly elliptic operators with non strongly convex symbols

TL;DR

This paper derives heat kernel estimates for a class of fourth order non-uniformly elliptic operators in two dimensions with non-strongly convex symbols, addressing challenges due to the lack of exponential constants.

Contribution

It introduces new heat kernel estimates for non-uniformly elliptic operators with non-strongly convex symbols, using sharp constants and Finsler-type distances.

Findings

Established heat kernel bounds with sharp constants

Developed estimates involving Finsler-type distances

Addressed difficulties due to non-strong convexity

Abstract

We obtain heat kernel estimates for a class of fourth order non-uniformly elliptic operators in two dimensions. Contrary to existing results, the operators considered have symbols that are not strongly convex. This rises certain difficulties as it is known that, as opposed to the strongly convex case, there is no absolute exponential constant. Our estimates involve sharp constants and Finsler-type distances that are induced by the operator symbol. The main result is based on two general hypotheses, a weighted Sobolev inequalitry and an interpolation inequality, which are related to the singularity or degeneracy of the coefficients.