# Boundary value problems for general first-order elliptic differential operators

## TL;DR

This paper develops a comprehensive framework for boundary value problems involving first-order elliptic differential operators on manifolds with boundary, accommodating non-selfadjoint boundary operators and non-pseudo-local conditions, and proves regularity and Fredholm properties.

## Contribution

It introduces a generalized approach to elliptic boundary conditions, unifying various existing conditions and proving regularity and Fredholm properties for solutions.

## Key findings

- Boundary conditions characterized in multiple equivalent ways
- Solutions exhibit regularity up to the boundary
- Imposing elliptic boundary conditions yields Fredholm operators on compact manifolds

## Abstract

We study boundary value problems for first-order elliptic differential operators on manifolds with compact boundary. The adapted boundary operator need not be selfadjoint and the boundary condition need not be pseudo-local. We show the equivalence of various characterisations of elliptic boundary conditions and demonstrate how the boundary conditions traditionally considered in the literature fit in our framework. The regularity of the solutions up to the boundary is proven. We show that imposing elliptic boundary conditions yields a Fredholm operator if the manifold is compact. We provide examples which are conveniently treated by our methods.