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

TY - JOUR

T1 - Krein-like extensions and the lower boundedness problem for elliptic operators on exterior domains

AU - Grubb, Gerd

PY - 2012/1/15

Y1 - 2012/1/15

N2 - For selfadjoint extensions A~ of a symmetric densely defined positive operator Amin, the lower boundedness problem is the question of whether A~ is lower bounded if and only if an associated operator T in abstract boundary spaces is lower bounded. It holds when the Friedrichs extension Aγ has compact inverse (Grubb, 1974, also Gorbachuk and Mikhailets, 1976); this applies to elliptic operators A on bounded domains.For exterior domains, Aγ-1 is not compact, and whereas the lower bounds satisfy m(T)≥m(A~), the implication of lower boundedness from T to A~ has only been known when m(T)>-m(Aγ). We now show it for general T.The operator Aa corresponding to T=aI, generalizing the Krein-von Neumann extension A0, 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;Aa)=cAtn/2m+O(t(n-1+ε)/2m) for t→∞.

AB - For selfadjoint extensions A~ of a symmetric densely defined positive operator Amin, the lower boundedness problem is the question of whether A~ is lower bounded if and only if an associated operator T in abstract boundary spaces is lower bounded. It holds when the Friedrichs extension Aγ has compact inverse (Grubb, 1974, also Gorbachuk and Mikhailets, 1976); this applies to elliptic operators A on bounded domains.For exterior domains, Aγ-1 is not compact, and whereas the lower bounds satisfy m(T)≥m(A~), the implication of lower boundedness from T to A~ has only been known when m(T)>-m(Aγ). We now show it for general T.The operator Aa corresponding to T=aI, generalizing the Krein-von Neumann extension A0, 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;Aa)=cAtn/2m+O(t(n-1+ε)/2m) for t→∞.

U2 - 10.1016/j.jde.2011.09.037

DO - 10.1016/j.jde.2011.09.037

M3 - Journal article

SN - 0022-0396

VL - 252

SP - 852-885.

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 2

ER -