Fractal curvatures and short-time asymptotics of heat content

TL;DR

This paper explores the short-time behavior of heat content in domains with smooth and fractal boundaries, introducing new mathematical insights into fractal curvatures and their potential role in heat asymptotics.

Contribution

It presents novel mathematical developments on heat content asymptotics for fractal boundaries and proposes the emergence of fractal curvatures in these asymptotics.

Findings

New results on heat content asymptotics for smooth and fractal boundaries

Introduction of fractal curvatures and their scaling properties

Conjecture linking fractal curvatures to heat content behavior

Abstract

The aim of our paper is twofold. First, we present new mathematical developments on the analysis of de Gennes' hypothesis on the short-time asymptotics of the heat content for bounded domains with smooth boundary and with fractal boundary. Second, we discuss new findings and concepts related to fractal curvatures for domains with fractal boundary. We conjecture that fractal curvatures and their scaling exponents will emerge in the short-time heat content asymptotics of domains with fractal boundary and the results discussed here are small initial contributions towards a resolution.