On the heat kernel of a class of fourth order operators in two dimensions: sharp Gaussian estimates and short time asymptotics

TL;DR

This paper investigates the heat kernel of certain fourth order elliptic operators in two dimensions, providing sharp Gaussian bounds and short time asymptotics without requiring strong convexity of the symbol.

Contribution

It offers new Gaussian estimates with optimal constants and short time asymptotics for operators lacking strong convexity assumptions.

Findings

Gaussian estimates with best constants for $L^{ abla}$ coefficients

Short time asymptotics for constant coefficient operators

Distinct behavior from strongly convex case

Abstract

We consider a class of fourth order uniformly elliptic operators in planar Euclidean domains and study the associated heat kernel. For operators with $L^{\infty}$ coefficients we obtain Gaussian estimates with best constants, while for operators with constant coefficients we obtain short time asymptotic estimates. The novelty of this work is that we do not assume that the associated symbol is strongly convex. The short time asymptotics reveal a behavior which is qualitatively different from that of the strongly convex case.