Boundary value problems for Dirac--type equations, with applications

Abstract

We prove regularity for a class of boundary value problems for first order elliptic systems, with boundary conditions determined by spectral decompositions, under coefficient differentiability conditions weaker than previously known. We establish Fredholm properties for Dirac-type equations with these boundary conditions. Our results include sharp solvability criteria, over both compact and non-compact manifolds; weighted Poincare and Schroedinger-Lichnerowicz inequalities provide asymptotic control in the non-compact case. One application yields existence of solutions for the Witten equation with a spectral boundary condition used by Herzlich in his proof of a geometric lower bound for the ADM mass of asymptotically flat 3-manifolds.

86 A4 pages, various style files

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Cited by in corpus (9)

• The Dirac system on the Anti-de Sitter Universe
• Global and uniqueness properties of stationary and static spacetimes with outer trapped surfaces
• Reduction Arguments for Geometric Inequalities Associated With Asymptotically Hyperboloidal Slices
• A positive mass theorem for low-regularity Riemannian metrics
• The Dirac operator on untrapped surfaces
• The mass of an asymptotically hyperbolic end and distance estimates
• The Conformal Flow of Metrics and the General Penrose Inequality
• The positive mass theorem and distance estimates in the spin setting
• Positive mass theorems for spin initial data sets with arbitrary ends and dominant energy shields