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

TL;DR

This paper establishes regularity and Fredholm properties for boundary value problems of Dirac-type equations with spectral boundary conditions, broadening applicability to compact and non-compact manifolds, and demonstrates an application to geometric analysis.

Contribution

It introduces weaker coefficient differentiability conditions for boundary value problems of Dirac-type equations and proves their regularity and Fredholm properties, including an application to the Witten equation.

Findings

Proved regularity under weaker differentiability conditions.

Established Fredholm properties for Dirac-type equations with spectral boundary conditions.

Applied results to existence of solutions for the Witten equation in geometric analysis.

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.