Krein-like extensions and the lower boundedness problem for elliptic operators

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:252

Article

Grubb, Gerd¹ 

[1] Copenhagen Univ Math Dept, DK-2100 Copenhagen, Denmark

关键词: Extension theory;    Elliptic operators;    Unbounded domains;    Lower bounds;    Pseudodifferential boundary operators;    Singular Green operators;    Eigenvalue asymptotics;    Perturbation theory;   

DOI  :  10.1016/j.jde.2011.09.037

来源: Elsevier

【 摘 要 】

For selfadjoint extensions (A) over tilde of a symmetric densely defined positive operator A(min), the lower boundedness problem is the question of whether (A) over tilde is lower bounded 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 and 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) >= m((A) over tilde), the implication of lower boundedness from T to (A) over tilde 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(A)t(n/2m) + O(t((n-1+epsilon)/2m)) for t -> infinity. (C) 2011 Elsevier Inc. All rights reserved.

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