  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.

Analysis of PDEs;Differential Geometry;Functional Analysis;Spectral Theory