# Boundary integral operators for the heat equation in time-dependent domains

Brügger, Rahel and Harbrecht, Helmut and Tausch, Johannes. (2020) Boundary integral operators for the heat equation in time-dependent domains. Preprints Fachbereich Mathematik, 2020 (02).

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Official URL: https://edoc.unibas.ch/79039/

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 $C^2$-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 by mimicking the proofs of [M. Costabel, Boundary integral operators for the heat equation, Integral Equations and Operator Theory, 13(4):498-552, 1990]. 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.