## 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 $\mathrm{\mathbf{c}\mathbf{u}\mathbf{r}\mathbf{l}}\phantom{\rule{0.166667em}{0ex}}\mathrm{\mathbf{c}\mathbf{u}\mathbf{r}\mathbf{l}}\phantom{\rule{0.166667em}{0ex}}\mathbf{u}-\eta \nabla \mathrm{div}\phantom{\rule{0.166667em}{0ex}}\mathbf{u}-{\kappa}^{2}\mathbf{u}=0$,
$\eta >0,\mathrm{Im}\phantom{\rule{0.166667em}{0ex}}{\kappa}^{2}\ge 0$, 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 $\kappa =0$.