Articles

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 I_H 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 S_K (i.e., the Krein-von Neumann extension of S),

S_K 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 = S_F^-1 (-Δ) v,   v = λ^-1(-Δ) u, with S_F 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 H_K,Ω, the Krein-von Neumann extension of the perturbed Laplacian -Δ+V (in short, the perturbed Krein Laplacian) defined on C^∞₀(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 C^1,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 | λ_K,Ω,j≤λ} = (2π)^-n v_n |Ω| λ^n/2+O(λ^(n-(1/2))/2)   as   λ→∞, where v_n=π^n/2/ Γ((n/2)+1) denotes the volume of the unit ball in ℝⁿ, |Ω denotes the volume of Ω, and λ_K,Ω,j, j∈ℕ, are the non-zero eigenvalues of H_K,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 C^∞₀(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 B_2,2 (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

TeX file (325k) or pdf file (789k)