Short time heat diffusion in compact domains with discontinuous transmission boundary conditions
Résumé
We consider a heat problem with discontinuous diffusion coefficients
and discontinuous transmission boundary conditions with a resistance
coefficient. For all compact $(\epsilon,\delta)$-domains
$\Omega\subset\mathbb{R}^n$ with a $d$-set boundary (for instance, a
self-similar fractal), we find the first term of the small-time
asymptotic expansion of the heat content in the complement of
$\Omega$, and also the second-order term in the case of a regular
boundary. The asymptotic expansion is different for the cases of
finite and infinite resistance of the boundary. The derived formulas
relate the heat content to the volume of the interior Minkowski
sausage and present a mathematical justification to the de Gennes'
approach. The accuracy of the analytical results is illustrated by
solving the heat problem on prefractal domains by a finite elements
method.
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