Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A hitchhiker's guide to first-order elliptic boundary value problems

Christian Baer, Lashi Bandara

TL;DR

This work provides a comprehensive, user-friendly operator-theoretic framework for first-order elliptic boundary value problems on manifolds with compact boundary. It introduces $\infty$-elliptic boundary conditions, adapted boundary operators, and a robust trace/regularity theory that yields interior and boundary regularity, as well as Fredholmness under coercivity at infinity. A deformation argument shows index invariance relative to boundary-condition modifications, and a matching-boundary construction yields a general relative index theorem linking global indices to local index densities. By unifying pseudo-local, local, APS, and absolute/relative boundary conditions, the paper broadens the toolkit for geometric and topological applications, including non-self-adjoint and non-Dirac-type operators like the Rarita-Schwinger operator. The results empower geometry and topology with a flexible, rigorous operator framework suited to a wide class of first-order elliptic problems.

Abstract

To empower the mathematical hitchhiker wishing to use operator methods in geometry and topology, we present this user's guide to first-order elliptic boundary value problems. Existence, regularity, and Fredholmness are discussed for general first-order elliptic operators on manifolds with compact boundary. The focus is on a very general class of elliptic boundary conditions, which contain those that are pseudo-local as a special case, yielding the relative index theorem. A new characterisation of a subclass of elliptic boundary conditions is also given.

A hitchhiker's guide to first-order elliptic boundary value problems

TL;DR

This work provides a comprehensive, user-friendly operator-theoretic framework for first-order elliptic boundary value problems on manifolds with compact boundary. It introduces -elliptic boundary conditions, adapted boundary operators, and a robust trace/regularity theory that yields interior and boundary regularity, as well as Fredholmness under coercivity at infinity. A deformation argument shows index invariance relative to boundary-condition modifications, and a matching-boundary construction yields a general relative index theorem linking global indices to local index densities. By unifying pseudo-local, local, APS, and absolute/relative boundary conditions, the paper broadens the toolkit for geometric and topological applications, including non-self-adjoint and non-Dirac-type operators like the Rarita-Schwinger operator. The results empower geometry and topology with a flexible, rigorous operator framework suited to a wide class of first-order elliptic problems.

Abstract

To empower the mathematical hitchhiker wishing to use operator methods in geometry and topology, we present this user's guide to first-order elliptic boundary value problems. Existence, regularity, and Fredholmness are discussed for general first-order elliptic operators on manifolds with compact boundary. The focus is on a very general class of elliptic boundary conditions, which contain those that are pseudo-local as a special case, yielding the relative index theorem. A new characterisation of a subclass of elliptic boundary conditions is also given.
Paper Structure (9 sections, 13 theorems, 68 equations)

This paper contains 9 sections, 13 theorems, 68 equations.

Table of Contents

  1. Setup and preliminaries
  2. Verifying the completeness assumption \ref{['Hyp.Complete']}
  3. Elliptic boundary conditions
  4. Adapted boundary operators
  5. General theory of elliptic boundary conditions
  6. Relation to the classical treatment of boundary conditions
  7. Fredholmness
  8. The matching boundary condition and relative index theory
  9. Note on higher regularity

Key Result

Theorem 1

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

Theorems & Definitions (39)

  • Example 1
  • Example 2
  • Theorem 1: The trace theorem BBan*Thm. 2.3 (i) and (ii)
  • Theorem 2: BBan2*Thm. 2.1
  • Remark 1
  • Example 3
  • Example 4
  • Remark 2
  • Theorem 3: BBan*Thm. 2.4
  • Definition 1: Adapted boundary operator
  • ...and 29 more