Effective Boundary Conditions for Heat Equation Arising from Anisotropic and Optimally Aligned Coatings in Three Dimensions

TL;DR

The paper investigates the heat equation in a 3D domain with a thin anisotropic coating that is optimally aligned with respect to the interface. By formulating weak solutions and deriving robust energy estimates, it proves compactness and identifies the limiting effective boundary conditions as the coating thickness vanishes. For Type I (equal tangent eigenvalues) and Type II (distinct tangent eigenvalues) conductivities, it derives a spectrum of EBCs that include local flux balances, Dirichlet-to-Neumann maps, and fractional Laplacian-type relations on the interface Γ, with detailed asymptotic regimes governed by scaling parameters $(\sigma,\mu,\mu_1,\mu_2)$ and thickness $\delta$. The results generalize prior 2D work to 3D and provide a rigorous multiscale framework for incorporating optimally aligned coatings into numerical and analytical models, facilitating accurate boundary modeling in applications like thermal barrier coatings and cell membranes. The work combines curvilinear coordinates, harmonic extensions, and spectral decompositions on $\Gamma$ to obtain explicit nonlocal boundary operators that govern heat transfer across the thin layer.

Abstract

We discuss the initial boundary value problem for a heat equation in a domain surrounded by a layer. The main features of this problem are twofold: on one hand, the layer is thin compared to the scale of the domain, and on the other hand, the thermal conductivity of the layer is drastically different from that of the bulk; moreover, the bulk is isotropic, but the layer is anisotropic and ``optimally aligned" in the sense that any vector in the layer normal to the interface is an eigenvector of the thermal tensor. We study the effects of the layer by thinking of it as a thickless surface, on which ``effective boundary conditions" (EBCs) are imposed. In the three-dimensional case, we obtain EBCs by investigating the limiting solution of the initial boundary value problem subject to either Dirichlet or Neumann boundary conditions as the thickness of the layer shrinks to zero. These EBCs contain not only the standard boundary conditions but also some nonlocal ones, including the Dirichlet-to-Neumann mapping and the fractional Laplacian. One of the main features of this work is to allow the drastic difference in the thermal conductivity in the normal direction and two tangential directions within the layer.

Theorems & Definitions (6)

• proof
• proof
• proof : Proof of the theorem \ref{['cpt']}
• proof
• proof : Proof of Theorem \ref{['thm']}
• proof : The proof of Theorem \ref{['thm2']}