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First-order elliptic boundary value problems on manifolds with non-compact boundary

Christian Baer, Lashi Bandara

TL;DR

This work develops a comprehensive framework for first-order elliptic boundary value problems on manifolds with smooth, potentially noncompact boundary. It constructs regularity for the maximal domain, a trace theorem onto adapted boundary data spaces, and a versatile boundary-value theory that includes APS-type and local boundary conditions. By introducing a geometric setup with an adapted boundary operator A and leveraging H^ inity functional calculus (or spectral calculus for A in the noncompact case), the authors obtain semi-Fredholm and Fredholm extensions under coercivity-type conditions, even when the boundary is noncompact. The theory is then specialized to Dirac-type operators, including twisted spinors and Callias-type perturbations, yielding Fredholmness criteria and connections to index theory in noncompact settings. The results offer a robust toolkit for boundary problems in geometric analysis and mathematical physics on manifolds with infinity-adjacent boundaries, broadening applicability beyond the compact-boundary paradigm.

Abstract

We consider first-order elliptic differential operators acting on vector bundles over smooth manifolds with smooth boundary, which is permitted to be noncompact. Under very mild assumptions, we obtain a regularity theory for sections in the maximal domain. Under additional geometric assumptions, and assumptions on an adapted boundary operator, we obtain a trace theorem on the maximal domain. This allows us to systematically study both local and nonlocal boundary conditions. In particular, the Atiyah-Patodi-Singer boundary condition occurs as a special case. Furthermore, we study contexts which induce semi-Fredholm and Fredholm extensions.

First-order elliptic boundary value problems on manifolds with non-compact boundary

TL;DR

This work develops a comprehensive framework for first-order elliptic boundary value problems on manifolds with smooth, potentially noncompact boundary. It constructs regularity for the maximal domain, a trace theorem onto adapted boundary data spaces, and a versatile boundary-value theory that includes APS-type and local boundary conditions. By introducing a geometric setup with an adapted boundary operator A and leveraging H^ inity functional calculus (or spectral calculus for A in the noncompact case), the authors obtain semi-Fredholm and Fredholm extensions under coercivity-type conditions, even when the boundary is noncompact. The theory is then specialized to Dirac-type operators, including twisted spinors and Callias-type perturbations, yielding Fredholmness criteria and connections to index theory in noncompact settings. The results offer a robust toolkit for boundary problems in geometric analysis and mathematical physics on manifolds with infinity-adjacent boundaries, broadening applicability beyond the compact-boundary paradigm.

Abstract

We consider first-order elliptic differential operators acting on vector bundles over smooth manifolds with smooth boundary, which is permitted to be noncompact. Under very mild assumptions, we obtain a regularity theory for sections in the maximal domain. Under additional geometric assumptions, and assumptions on an adapted boundary operator, we obtain a trace theorem on the maximal domain. This allows us to systematically study both local and nonlocal boundary conditions. In particular, the Atiyah-Patodi-Singer boundary condition occurs as a special case. Furthermore, we study contexts which induce semi-Fredholm and Fredholm extensions.
Paper Structure (26 sections, 67 theorems, 245 equations, 1 figure)

This paper contains 26 sections, 67 theorems, 245 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setup and statement of main results
  3. Notation
  4. Minimal setup and regularity
  5. Geometric setup and existence
  6. Sufficient criterion for completeness
  7. The first trace theorem and regularity
  8. The model operator
  9. The maximal domain
  10. Boundary value problems
  11. Boundary conditions
  12. Regularity revisited
  13. Boundary conditions via projectors
  14. Atiyah-Patodi-Singer boundary condition
  15. Local boundary conditions
  16. ...and 11 more sections

Key Result

Theorem 2.1

Under the assumptions Hyp:StdFirst--Hyp:StdLast, the space ${\rm C}^{\infty}_{\rm c}(M;E)$ is dense in $\mathrm{dom}(D_{\max})$ with respect to the corresponding graph norm. Moreover, the restriction map to the boundary has a unique bounded extension

Figures (1)

  • Figure 1: Choice of neighbourhoods

Theorems & Definitions (163)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Definition 2.6: Boundary condition, semi-regularity, regularity
  • Theorem 2.7
  • Corollary 2.8
  • Theorem 2.9
  • Remark 2.10
  • ...and 153 more