First order operators and boundary triples

TL;DR

This paper introduces a first order approach to boundary triples for Laplace operators, enabling intrinsic boundary definitions and applications to manifolds with boundary where traditional methods fail.

Contribution

It develops a novel first order framework for boundary triples, allowing intrinsic boundary operators and norms, and connects to Dirac operators on manifolds.

Findings

First order approach extends boundary triples to manifolds with boundary.

Intrinsic Dirichlet-to-Neumann map defined within the new framework.

Connection established between first order boundary triples and Dirac operators.

Abstract

The aim of the present paper is to introduce a first order approach to the abstract concept of boundary triples for Laplace operators. Our main application is the Laplace operator on a manifold with boundary; a case in which the ordinary concept of boundary triples does not apply directly. In our first order approach, we show that we can use the usual boundary operators also in the abstract Green's formula. Another motivation for the first order approach is to give an intrinsic definition of the Dirichlet-to-Neumann map and intrinsic norms on the corresponding boundary spaces. We also show how the first order boundary triples can be used to define a usual boundary triple leading to a Dirac operator.