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

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 we establish that the obtained Fredholm of index zero. Their finite dimensional kernels characterized show their dimension is equal to number topological invariants domain's boundary, other words sum its Betti numbers. This explained by fundamental discovery associated bilinear forms agree with those induced 2D surface H-1/2 de Rham Hilbert complexes whose underlying inner-products non-local inner products defined through classical single-layer Laplacian. Decay conditions well-posedness natural energy spaces system unbounded exterior also presented.