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First-Kind Boundary Integral Equations for the Dirac Operator in 3d Lipschitz Domains

by E. Schulz and R. Hiptmair

(Report number 2020-69)

Abstract
We develop novel first-kind boundary integral equations for Euclidean Dirac operators in 3D Lipschitz domains comprising square-integrable potentials and involving only weakly singular kernels. Generalized Garding inequalities are derived and we establish that the obtained boundary integral operators are Fredholm of index zero. Their finite dimensional kernels are characterized and we show that their dimension is equal to the number of topological invariants of the domain’s boundary, in other words to the sum of its Betti numbers. This is explained by the fundamental discovery that the associated bilinear forms agree with those induced by the 2D surface Dirac operators for H−1/2 surface de Rham Hilbert complexes whose underlying inner-products are the non-local inner products defined through the classical single-layer boundary integral operators for the Laplacian. Decay conditions for well-posedness in natural energy spaces of the Dirac system in unbounded exterior domains are also presented.

Keywords: Dirac, Hodge-Dirac, potential representation, representation formula, jump relations, first-kind boundary integral operators, boundary integral equations, surface Dirac operators, coercivity

@Techreport{SH20_942,
  author = {E. Schulz and R. Hiptmair},
  title = {First-Kind Boundary Integral Equations for the Dirac Operator in 3d Lipschitz Domains},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2020-69},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-69.pdf },
  year = {2020}
}

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