Effective Boundary Conditions on Coated Bodies.

摘要

The goal of this work is to study how the physical properties of a thin coating of insulation, such as thermal conductivity and thickness, determine its overall effect on an object. Particular attention will be paid to the case where the coating exhibits anisotropic, or direction-dependent, thermal conductivity. Two general questions will be considered: how the properties of the coating determine the thermal protection of the insulated body, and which boundary conditions appear in a limiting boundary value problem on the interior body. The first question will be addressed via explicit estimates that compare the true temperature of the body to the temperature after assuming perfect insulation. This will be accomplished by analyzing the weak formulation of the heat equation and applying the method of eigenexpansion. The second question will be addressed by introducing auxiliary harmonic functions into the weak formulation of the boundary value problem. In this question we will restrict ourselves to objects in the plane and coatings that are optimally aligned.