First-Kind Boundary Integral Equations for the Hodge-Helmholtz Equation
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2017-04
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Report
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Abstract
We adapt the variational approach to the analysis of first-kind boundary integral equations associated with strongly elliptic partial differential operators from [{ M.~Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal., 19 (1988), pp. 613-626.] to the (scaled) Hodge-Helmholtz equation curlcurlu−η∇divu−κ2u=0, η>0,Imκ2≥0, on Lipschitz domains in 3D Euclidean space, supplemented with natural complementary boundary conditions, which, however, fail to bring about strong ellipticity. Nevertheless, a boundary integral representation formula can be found, from which we can derive boundary integral operators. They induce bounded and coercive sesqui-linear forms in the natural energy trace spaces for the \HH equation. We can establish precise conditions on η,κ that guarantee unique solvability of the two first-kind boundary integral equations associated with the natural boundary value problems for the Hodge-Helmholtz equations. Particular attention will be given to the case κ=0.
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published
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2017-22
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Seminar for Applied Mathematics, ETH Zurich
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Maxwell’s Equations; static limit; Hodge-Laplacian; potential representations; jump relations; first-kind boundary integral equations; coercive integral equations
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03632 - Hiptmair, Ralf / Hiptmair, Ralf