Multi-parameter perturbations for the space-periodic heat equation

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 domain and we show that the solution depends smoothly on the shape of the transmission interface, boundary data, and transmission parameters. Finally, in the last part of the paper, we fix all parameters except for the transmission parameters and outline a strategy to deduce an explicit expansion of the solution using Neumann-type series.