Gradient and Transport Estimates for Heat Flow on Nonconvex Domains

TL;DR

This paper establishes sharp gradient and transport estimates for heat flow with Neumann boundary conditions on nonconvex domains, revealing a novel dependence and characterizing boundary curvature bounds.

Contribution

It introduces new sharp gradient and transport estimates for heat flow on nonconvex domains, with a novel  dependence, and characterizes boundary curvature bounds via these estimates.

Findings

Derived sharp gradient estimates with  dependence.

Provided an equivalent characterization of boundary second fundamental form bounds.

Established quantitative bounds relating curvature and heat flow behavior.

Abstract

For the Neumann heat flow on nonconvex Riemannian domains $D\subset M$, we provide sharp gradient estimates and transport estimates with a novel $\sqrt t$-dependence, for instance, $$\text{Lip}( P^D_tf)\le e^{2S \, \sqrt{t/\pi}+\mathcal{O}(t)}\cdot \text{Lip} (f),$$ and we provide an equivalent characterization of the lower bound $S$ on the second fundamental form of the boundary in terms of these quantitative estimates.