Short time heat diffusion in compact domains with discontinuous transmission boundary conditions

TL;DR

This paper derives the small-time asymptotic expansion of heat content in domains with fractal boundaries and discontinuous transmission conditions, linking it to Minkowski content and validating with finite element simulations.

Contribution

It provides new formulas for heat content asymptotics in domains with complex fractal boundaries and discontinuous boundary conditions, including resistance effects.

Findings

Asymptotic expansion formulas for heat content in fractal domains.

Relation of heat content to interior Minkowski sausage.

Validation through finite element simulations on prefractal domains.

Abstract

We consider a heat problem with discontinuous diffusion coefficientsand discontinuous transmission boundary conditions with a resistancecoefficient. For all compact $(\epsilon,\delta)$-domains $\Omega\subset\mathbb{R}^n$ with a $d$-set boundary (for instance, aself-similar fractal), we find the first term of the small-timeasymptotic expansion of the heat content in the complement of$\Omega$, and also the second-order term in the case of a regularboundary. The asymptotic expansion is different for the cases offinite and infinite resistance of the boundary. The derived formulasrelate the heat content to the volume of the interior Minkowskisausage and present a mathematical justification to the de Gennes'approach. The accuracy of the analytical results is illustrated bysolving the heat problem on prefractal domains by a finite elementsmethod.