First-Kind Boundary Integral Equations for the Hodge-Helmholtz Operator
Résumé
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 curl curl u − η∇div u − κ 2 u = 0, η > 0, Im κ 2 ≥ 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 Hodge-Helmholtz 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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