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## 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 $\left[$M. Costabel , Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal., 19 (1988), pp. 613–626.$\right]$ to the (scaled) Hodge-Helmholtz equation $\mathrm{𝐜𝐮𝐫𝐥}\phantom{\rule{0.166667em}{0ex}}\mathrm{𝐜𝐮𝐫𝐥}\phantom{\rule{0.166667em}{0ex}}𝐮-\eta \nabla \mathrm{div}\phantom{\rule{0.166667em}{0ex}}𝐮-{\kappa }^{2}𝐮=0$, $\eta >0,\mathrm{Im}\phantom{\rule{0.166667em}{0ex}}{\kappa }^{2}\ge 0$, 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 $\eta ,\kappa$ 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 needs to be given to the case $\kappa =0$.