in Mathematical Physics, Spectral Theory and Stochastic Analysis, M. Demuth and W. Kirsch (eds.), 1-106, Oper. Theory Adv. Appl. 232, 2013 [DOI: 10.1007/978-3-0348-0591-9_1]

A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains

Mark S. Ashbaugh, Fritz Gesztesy, Marius Mitrea, Roman Shterenberg, and Gerald Teschl

In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein-von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, S≥ ε IH for some ε >0 in a Hilbert space H to an abstract buckling problem operator.

In the concrete case where S=-Δ|C0(Ω) in L2(Ω; dn x) for Ω⊂ℝn an open, bounded (and sufficiently regular) set, this recovers, as a particular case of a general result due to G. Grubb, that the eigenvalue problem for the Krein Laplacian SK (i.e., the Krein-von Neumann extension of S),

SK v = λ v,   λ ≠ 0,
is in one-to-one correspondence with the problem of the buckling of a clamped plate,
(-Δ)2u=λ (-Δ) u   in   Ω,   λ ≠ 0,   u∈ H02(Ω),
where u and v are related via the pair of formulas
u = SF-1 (-Δ) v,   v = λ-1(-Δ) u,
with SF the Friedrichs extension of S.

This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.).

In the second, and principal part of this survey, we study spectral properties for HK,Ω, the Krein-von Neumann extension of the perturbed Laplacian -Δ+V (in short, the perturbed Krein Laplacian) defined on C0(Ω), where V is measurable, bounded and nonnegative, in a bounded open set Ω⊂ℝn belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class C1,r, r>1/2. (Contrary to other uses of the notion of “domain”, a domain in this survey denotes an open set without any connectivity hypotheses. In addition, by a “smooth domain” we mean a domain with a sufficiently smooth, typically, a C-smooth, boundary.) In particular, in the aforementioned context we establish the Weyl asymptotic formula

#{j∈ℕ | λK,Ω,j≤λ} = (2π)-n vn |Ω| λn/2+O(λ(n-(1/2))/2)   as   λ→∞,
where vnn/2/ Γ((n/2)+1) denotes the volume of the unit ball in n, denotes the volume of Ω, and λK,Ω,j, j∈ℕ, are the non-zero eigenvalues of HK,Ω, listed in increasing order according to their multiplicities. We prove this formula by showing that the perturbed Krein Laplacian (i.e., the Krein-von Neumann extension of -Δ+V defined on C0(Ω)) is spectrally equivalent to the buckling of a clamped plate problem, and using an abstract result of Kozlov from the mid 1980's. Our work builds on that of Grubb in the early 1980's, who has considered similar issues for elliptic operators in smooth domains, and shows that the question posed by Alonso and Simon in 1980 pertaining to the validity of the above Weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting.

We also study certain exterior-type domains Ω = ℝn∖ K, n≥ 3, with K⊂ ℝn compact and vanishing Bessel capacity B2,2 (K) = 0, to prove equality of Friedrichs and Krein Laplacians in L2(Ω; dn x), that is, -Δ|C0(Ω) has a unique nonnegative self-adjoint extension in L2(Ω; dn x).

MSC2000: Primary 35J25, 35J40, 35P15; Secondary 35P05, 46E35, 47A10, 47F05.
Keywords: Lipschitz domains, Krein Laplacian, eigenvalues, spectral analysis, Weyl asymptotics, buckling problem

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