Fractal curvatures and short-time asymptotics of heat content

Abstract: The aim of our paper is twofold. First, we present applied mathematics results on the heat content of 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.