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
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),
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 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 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
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).