Boundary integral exterior calculus
Year: 2025
Type: article
Abstract: We report a surprising and deep structural property of boundary integral operators occurring in first-kind boundary integral equations associated with Hodge–Dirac and Hodge–Laplace operators for de Rham Hilbert complexes on a bounded domain \Omega in a Riemannian manifold. We show that, as regards their induced bilinear forms, those boundary integral operators are Hodge–Dirac and Hodge–Laplace operators in the weak sense, this time set in a trace de Rham Hilbert complex on the boundary \partial\Omega whose underlying spaces of differential forms are equipped with non-local inner products defined through layer potentials. On the way to this main result we conduct a thorough analysis of layer potentials in operator-induced trace spaces and derive representation formulas.
DOI: 10.4171/jems/1621
Authors: Erick Schulz, Ralf Hiptmair, Stefan Kurz
Institution: ETH Zurich
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4408843271