Pointwise Lower bounds on the Heat Kernels of Uniformally Elliptic Operators in Bounded Regions

TL;DR

This paper establishes explicit pointwise lower bounds for heat kernels of higher order elliptic operators with Dirichlet boundary conditions in bounded regions, highlighting boundary decay without requiring smooth coefficients.

Contribution

It provides the first explicit lower bounds for heat kernels of higher order elliptic operators with minimal coefficient regularity.

Findings

Explicit boundary decay behavior of heat kernels

Lower bounds applicable to operators with bounded measurable coefficients

No smoothness assumptions needed on coefficients

Abstract

We obtain pointwise lower bounds for heat kernels of higher order differential operators with Dirichlet boundary conditions on bounded domains in $\R^N$. The bounds exhibit explicitly the nature of the spatial decay of the heat kernel close to the boundary. We make no smoothness assumptions on our operator coefficients which we assume only to be bounded and measurable.