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

TL;DR

This paper introduces new first-kind boundary integral equations for the 3D Dirac operator on Lipschitz domains, linking topological invariants to the kernel dimension and establishing Fredholm properties.

Contribution

It develops novel boundary integral equations for the 3D Dirac operator with weakly singular kernels, characterizes their kernels via topological invariants, and connects them to surface Dirac operators.

Findings

Boundary integral operators are Fredholm of index zero.

Kernel dimension equals the sum of Betti numbers of the boundary.

Decay conditions for well-posedness in exterior domains are provided.

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.