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First-Kind Boundary Integral Equations for the Hodge-Helmholtz Equation

by X. Claeys and R. Hiptmair

(Report number 2017-22)

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 \({\bf curl}\,{\bf curl}\,\mathbf{u}-\eta\nabla\mathrm{div}\,\mathbf{u}-\kappa^{2}\mathbf{u}=0\), \(\eta>0, \mathrm{Im}\,\kappa^{2}\geq 0\), 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 \(\eta,\kappa\) 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 \(\kappa=0\).

Keywords: Maxwell's Equations; static limit, Hodge-Laplacian; potential representations, jump relations, first-kind boundary integral equations; coercive integral equations.

@Techreport{CH17_718,
  author = {X. Claeys and R. Hiptmair},
  title = {First-Kind Boundary Integral Equations for the Hodge-Helmholtz Equation},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2017-22},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2017/2017-22.pdf },
  year = {2017}
}

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