Boundary Integral Operators for the Heat Equation in Time-Dependent Domains
Rahel Brügger, Helmut Harbrecht, Johannes Tausch · Integral Equations and Operator Theory · 2022
Abstract This article provides a functional analytical framework for boundary integral equations of the heat equation in time-dependent domains. More specifically, we consider a non-cylindrical domain in space-time that is the $$C^2$$ C 2 -diffeomorphic image of a cylinder, i.e., the tensor product of a time interval and a fixed domain in space. On the non-cylindrical domain, we introduce Sobolev spaces, trace lemmata and provide the mapping properties of the layer operators. Here it is critical that the Neumann trace requires a correction term for the normal velocity of the moving boundary. Therefore, one has to analyze the situation carefully.