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

2025/10/19 by Christian Baer, Baer, Christian, Lashi Bandara +1
Mathematics · #35J56 #58J32 #Advanced Operator Algebra Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2510.16919

openalex publication_date 2025/10/19 · openalex created_date 2025/10/22 · openalex updated_date 2026/07/28

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.

Citations

• First-order elliptic boundary value problems on manifolds with non-compact boundary
• Regularity and index theory for Dirac-Schrödinger systems with Lipschitz coefficients
• Spectral asymmetry and Riemannian geometry. II

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• Boundary value problems for first order elliptic wedge operators
• Index defects in the theory of nonlocal boundary value problems and the eta invariant
• The homotopy classification and the index of boundary value problems for general elliptic operators
• Elliptic systems of variable order