2020/12/31 by Erick Schulz, Ralf Hiptmair
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Boundary (topology) #Differential Equations and Boundary Problems #Dirac (video compression format) #Dirac algebra #Dirac equation #Dirac operator #Integral equation #Jump #Lipschitz continuity #Mathematical analysis #Mathematical physics #Mathematics #Numerical methods in inverse problems #Operator (biology) #Pure mathematics #Representation (politics) #math.AP #math.DG #msc:31A10 #msc:34L40 #msc:35F15 #msc:35Q61 #msc:45A05 #msc:45E05 #msc:45P05
paper · pdf · doi:10.1137/20m1389224
published as SIAM Journal on Mathematical Analysis 54.1 (2022): 616-648 · 27 pages, 2 figures
arxiv created 2021/10/13 · openalex publication_date 2022/01/20 · openalex created_date 2022/01/25 · arxiv updated 2022/03/01 · openalex updated_date 2026/06/26
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.