Effective Boundary Conditions Arising from the Heat Equation with Three-dimensional Interior Inclusion

TL;DR

This paper derives effective boundary conditions for the heat equation in a domain with a thin, anisotropic layer of drastically different thermal conductivity, revealing complex boundary behaviors as the layer thickness approaches zero.

Contribution

It introduces a rigorous analysis of the limit behavior of the heat equation with a thin anisotropic layer, resulting in novel effective boundary conditions including nonstandard types.

Findings

Derivation of effective boundary conditions as the layer thickness tends to zero

Identification of nonstandard boundary conditions such as fractional Laplacian

Analysis of anisotropic effects on heat transfer at the boundary

Abstract

We study the initial boundary value problem for a heat equation in a domain containing a thin layer. The thermal conductivity of the layer is drastically different from that of the bulk of the domain; moreover, the layer is anisotropic and ``optimally aligned" in the sense that the normal direction in the layer is always an eigenvector of the thermal tensor. To reveal the effects of the layer, we regard it as a thickless surface on which ``effective boundary conditions" (EBCs) are satisfied by the limit of solutions of the initial boundary value problem as the thickness of the layer shrinks to zero. These EBCs are rich in variety and type, including some nonstandard ones such as the Dirichlet-to-Neumann mapping and the fractional Laplacian.