6 research outputs found
Krein-like extensions and the lower boundedness problem for elliptic operators
For selfadjoint extensions tilde-A of a symmetric densely defined positive
operator A_min, the lower boundedness problem is the question of whether
tilde-A is lower bounded {\it if and only if} an associated operator T in
abstract boundary spaces is lower bounded. It holds when the Friedrichs
extension A_gamma has compact inverse (Grubb 1974, also Gorbachuk-Mikhailets
1976); this applies to elliptic operators A on bounded domains.
For exterior domains, A_gamma ^{-1} is not compact, and whereas the lower
bounds satisfy m(T)\ge m(tilde-A), the implication of lower boundedness from T
to tilde-A has only been known when m(T)>-m(A_gamma). We now show it for
general T.
The operator A_a corresponding to T=aI, generalizing the Krein-von Neumann
extension A_0, appears here; its possible lower boundedness for all real a is
decisive. We study this Krein-like extension, showing for bounded domains that
the discrete eigenvalues satisfy
N_+(t;A_a)=c_At^{n/2m}+O(t^{(n-1+varepsilon)/2m}) for t\to\infty .Comment: 35 pages, revised for misprints and accepted for publication in
Journal of Differential Equation
Extension Theory and Krein-type Resolvent Formulas for Nonsmooth Boundary Value Problems
For a strongly elliptic second-order operator on a bounded domain
it has been known for many years how to interpret
the general closed -realizations of as representing boundary
conditions (generally nonlocal), when the domain and coefficients are smooth.
The purpose of the present paper is to extend this representation to nonsmooth
domains and coefficients, including the case of H\"older
-smoothness, in such a way that pseudodifferential
methods are still available for resolvent constructions and ellipticity
considerations. We show how it can be done for domains with
-smoothness and operators with -coefficients, for
suitable and . In particular, Kre\u\i{}n-type resolvent
formulas are established in such nonsmooth cases. Some unbounded domains are
allowed.Comment: 62 page