We are concerned here with
determining some closed operators associated with a given elliptic differential
operator A of order 2m and some in general nonlocal boundary operators. We seek
conditions in particular which guarantee that the result is a normally solvable
operator, i.e. with closed graph and closed range in the sense of Visik. We follow
basically the method used in Bade and Freeman in the sense that we regard the
operator with nonlocal boundary conditions as a perturbation of an operator
with boundary conditions defined by a normal set of differential operators
B = {B0,⋯,Bm−1} satisfying the condition of Agmon, Douglis and Nirenberg (also
Browder, and Schechter). Since the basic a priori estimate valid for such systems
essentially says that the resulting operator has closed graph we call such a system
(A,B) closable elliptic.
In addition to dealing with higher order elliptic operators and general boundary
conditions we also drop the requirement that our region be relatively compact and
instead make the weaker requirement that the differential operator in H2m(Ω) with
local boundary conditions yields an operator with closed range. We work here in L2
only and consider operators defined in H2m(Ω), in the graph topology associated
with the so called maximal operator and in a family of spaces interpolated
between these two. Most of our results can be obtained, at least for relatively
compact regions, in Lp with 1 < p < ∞ at the expense of a somewhat more
complicated treatment. A particular complication arises from the fact that different
interpolation methods which yield the same spaces in L2 do not in general in
Lp,p≠2.
