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.