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Boundary integral operators for the heat equation

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Abstract

We study the integral operators on the lateral boundary of a space-time cylinder that are given by the boundary values and the normal derivatives of the single and double layer potentials defined with the fundamental solution of the heat equation. For Lipschitz cylinders we show that the 2×2 matrix of these operators defines a bounded and positive definite bilinear form on certain anisotropic Sobolev spaces. By restriction, this implies the positivity of the single layer heat potential and of the normal derivative of the double layer heat potential. Continuity and bijectivity of these operators in a certain range of Sobolev spaces are also shown. As an application, we derive error estimates for various Galerkin methods. An example is the numerical approximation of an eddy current problem which is an interface problem with the heat equation in one domain and the Laplace equation in a second domain. Results of numerical computations for this problem are presented.

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Authors and Affiliations

  1. Technische Hochschule Darmstadt, Schloßgartenstraße 7, D-6100, Darmstadt, W. Germany

    Martin Costabel

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  1. Martin Costabel
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Parts of this work were done while the author had visiting positions at the Carnegie Mellon University, Pittsburgh, USA, and at the Université de Nantes, France, or was supported by the DFG-Forschergruppe KO 634/32-1.

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Costabel, M. Boundary integral operators for the heat equation. Integr equ oper theory 13, 498–552 (1990). https://doi.org/10.1007/BF01210400

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