Realisations of elliptic operators on compact manifolds with boundary

TL;DR

This paper characterizes boundary conditions for elliptic differential operators on manifolds with boundary, describing their boundary value spaces, and analyzing spectral properties, regularity, and well-posedness of the associated boundary value problems.

Contribution

It extends the Bär-Ballmann approach to higher order elliptic operators, providing a detailed description of boundary value spaces and conditions for Fredholm and regular realizations.

Findings

Boundary spaces described via Calderón projectors

Characterization of boundary conditions for Fredholm and regular realizations

Results on spectral theory and well-posedness for higher order problems

Abstract

This paper investigates realisations of elliptic differential operators of general order on manifolds with boundary following the approach of B\"ar-Ballmann to first order elliptic operators. The space of possible boundary values of elements in the maximal domain is described as a Hilbert space densely sandwiched between two mixed order Sobolev spaces. The description uses Calder\'on projectors which, in the first order case, is equivalent to results of B\"ar-Bandara using spectral projectors of an adapted boundary operator. Boundary conditions that induce Fredholm as well as regular realisations, and those that admit higher order regularity, are characterised. In addition, results concerning spectral theory, homotopy invariance of the Fredholm index, and well-posedness for higher order elliptic boundary value problems are proven.