First-Kind Boundary Integral Equations for the Hodge--Helmholtz Operator
Xavier Claeys, Ralf Hiptmair · SIAM Journal on Mathematical Analysis · 2019
We adapt the variational approach to the analysis of first-kind boundary integral equations associated with strongly elliptic partial differential operators from [M. Costabel, SIAM J. Math. Anal., 19 (1988), pp. 613--626] to the (scaled) Hodge--Helmholtz equation ${curl},{curl},{u}-\eta abla{div}\, {u}-\kappa^{2}{u}=0$, $\eta>0, {Im}\,\kappa^{2}\geq 0$, on Lipschitz domains in three-dimensional 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 sesquilinear forms in the natural energy trace spaces for the Hodge--Helmholtz 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 $\kappa=0$.