Overall Objectives
Research Program
Application Domains
Highlights of the Year
New Software and Platforms
New Results
Bilateral Contracts and Grants with Industry
Partnerships and Cooperations
XML PDF e-pub
PDF e-Pub

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 𝐜𝐮𝐫𝐥𝐜𝐮𝐫𝐥𝐮-η div 𝐮-κ2𝐮=0, η>0, Im κ20, 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 needs to be given to the case κ=0.