Gradient and Transport Estimates for Heat Flow on Nonconvex Domains
Karl-Theodor Sturm
TL;DR
This work establishes a sharp connection between boundary geometry and Neumann heat flow on nonconvex domains by proving the equivalence between a lower bound on the second fundamental form, $s\ge -S$, and quantitative gradient and transport estimates for the Neumann semigroup. The core method combines probabilistic representations via reflected Brownian motion with local time, Khasminskii's lemma, and a Kuwada-type duality to derive a novel $\sqrt{t}$-dependent exponent in the contraction and gradient bounds. The results extend the von Renesse–Sturm program to nonconvex domains and provide a geometric-analytic characterization of boundary curvature through semigroup estimates, including explicit asymptotics for the boundary local time $\mathbb E_x[\ell_t]$. Overall, the paper supplies sharp tools for controlling transport and gradient flows in rough geometric settings and highlights the intimate link between boundary geometry and heat-flow regularity.
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, $$\text{Lip}( P^D_tf)\le e^{2S \, \sqrt{t/π}+\mathcal{O}(t)}\cdot \text{Lip} (f),$$ 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.