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

We consider a heat problem with discontinuous diffusion coefficientsand discontinuous transmission boundary conditions with a resistancecoefficient. For all compact $(ϵ,\delta )$-domains $\mathrm{\Omega }\subset {\mathbb{R}}^n$ 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$\mathrm{\Omega }$, 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.