Hardy inequality and heat semigroup estimates for Riemannian manifolds with singular data

TL;DR

This paper derives upper bounds for heat content in Riemannian manifolds with boundary, utilizing Hardy inequalities and specific initial conditions, providing near-sharp estimates under certain geometric and spectral assumptions.

Contribution

It establishes new heat content bounds on Riemannian manifolds with singular data using Hardy inequalities, extending previous results to more general geometric settings.

Findings

Upper bounds are close to sharp under Hardy inequality conditions.

Bounds depend on initial data with singularities near the boundary.

Results apply to geodesically complete Riemannian manifolds.

Abstract

Upper bounds are obtained for the heat content of an open set D in a geodesically complete Riemannian manifold M with Dirichlet boundary condition on bd(D), and non-negative initial condition. We show that these upper bounds are close to being sharp if (i) the Dirichlet-Laplace-Beltrami operator acting in $L^2(D)$ satisfies a strong Hardy inequality with weight $r^2$, (ii) the initial temperature distribution, and the specific heat of D are given by $r^{-a}$ and $r^{-b}$ respectively, where $r$ is the distance to the boundary, and 1<a<2, 1<b<2.