Effective Boundary Conditions for the Heat Equation with Interior Inclusion (Communications in Mathematical Research, 2020)
Of concern is the scenario of a heat equation on a domain that contains a thin layer, on which the thermal conductivity is drastically differentfrom that in the bulk. The multi-scales in the spatial variable and the thermal conductivity lead to computational difficulties, so we may think of thethin layer as a thickless surface, on which we impose "effective boundary conditions" (EBCs). These boundary conditions not only ease the computationalburden, but also reveal the effect of the inclusion. In this paper, by consideringthe asymptotic behavior of the heat equation with interior inclusion subject toDirichlet boundary condition, as the thickness of the thin layer shrinks, we derive, on a closed curve inside a two-dimensional domain, EBCs which includea Poisson equation on the curve, and a non-local one. It turns out that theEBCs depend on the magnitude of the thermal conductivity in the thin layer,compared to the reciprocal of its thickness.
Keywords: Heat equation; effective boundary conditions; weak solution; a priori estimates; asymptotic behavior.