Fractal curvatures in short-time heat content asymptotics

Establish that Lipschitz–Killing fractal curvature measures and their associated scaling exponents appear in the short-time asymptotic expansion of the heat content N(t) for bounded domains Ω ⊂ R^n with fractal boundary ∂Ω, under appropriate geometric assumptions on ∂Ω.

Background

The paper studies the short-time behavior of heat content for domains with smooth and fractal boundaries and develops connections to curvature measures. For smooth boundaries, curvatures enter higher-order terms of classical asymptotic expansions; for fractal boundaries, the authors argue that analogues of curvature measures (fractal curvatures) should govern the short-time regime.

This conjecture aims to unify geometric and analytic perspectives by asserting that appropriate fractal curvatures (and their exponents) control leading-order and correction terms in heat content asymptotics for fractal interfaces, extending known results for Minkowski content and surface area in the smooth setting.