Multi-parameter perturbations for the space-periodic heat equation

TL;DR

This paper investigates how solutions to the space-periodic heat equation vary smoothly with multiple parameters, including shape, boundary data, and conductivity, providing explicit expansions for the solution.

Contribution

It establishes the smooth dependence of heat equation solutions on multiple parameters and develops a strategy for explicit solution expansion using Neumann series.

Findings

Solutions depend smoothly on shape, boundary data, and conductivity.

Explicit solution expansions can be derived using Neumann series.

Results apply to periodic two-phase composite materials.

Abstract

This paper is divided into three parts. The first part focuses on periodic layer heat potentials, demonstrating their smooth dependence on regular perturbations of the support of integration. In the second part, we present an application of the results from the first part. Specifically, we consider a transmission problem for the heat equation in a periodic two-phase composite material and we show that the solution depends smoothly on the shape of the transmission interface, boundary data, and conductivity parameters. Finally, in the last part of the paper, we fix all parameters except for the contrast parameter and outline a strategy to deduce an explicit expansion of the solution using a Neumann-type series.