Section: New Results
First kind boundary integral formulation for the Hodge-Helmholtz equation
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 , , 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 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 needs to be given to the case .