Mathematics – Analysis of PDEs
Scientific paper
2010-02-24
Mathematics
Analysis of PDEs
35 pages, revised for misprints and accepted for publication in Journal of Differential Equations
Scientific paper
10.1016/j.jde.2011.09.037
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 .
No associations
LandOfFree
Krein-like extensions and the lower boundedness problem for elliptic operators does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Krein-like extensions and the lower boundedness problem for elliptic operators, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Krein-like extensions and the lower boundedness problem for elliptic operators will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-260785