Spectral theory for perturbed Krein Laplacians in nonsmooth domains

2009

62 citations recorded by the source corpus.

Abstract We study spectral properties for H K , Ω , the Krein–von Neumann extension of the perturbed Laplacian − Δ + V defined on C 0 ∞ ( Ω ) , where V is measurable, bounded and nonnegative, in a bounded open set Ω ⊂ R n belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class C 1 , r , r > 1 / 2 . In particular, in the aforementioned context we establish the Weyl asymptotic formula # { j ∈ N | λ K , Ω , j ⩽ λ } = ( 2 π ) − 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 R n , 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 1980s. Our work builds on that of Grubb in the early 1980s, 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 = R n ∖ K , n ⩾ 3 , with K ⊂ R n compact and vanishing Bessel capacity B 2 , 2 ( K ) = 0 , to prove equality of Friedrichs and Krein Laplacians in L 2 ( Ω ; d n x ) , that is, − Δ | C 0 ∞ ( Ω ) has a unique nonnegative self-adjoint extension in L 2 ( Ω ; d n x ) .

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a560a9f8c05de77f4f729e5ee2938f455c1f708d

Citations in the corpus

• A description of all self-adjoint extensions of the Laplacian and Kreĭn-type resolvent formulas on non-smooth domains (2009, unknown): “For a variety of recent new results on SK in connection with Weyl-type spectral asymptotics for perturbed Laplacians on nonsmooth bounded open domains in R, and for a connection to an abstract buckling problem that illustrates the relevance of SK in elasticity theory, we refer to [19] and [20].”
• Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples (2012, unknown): “If η = 0, then the operator A1 is the Krein–von Neumann or “soft” extension of S, which was studied, e.g. in [Ashbaugh et al., 2010; Behrndt and Langer, 2007; Everitt and Markus, 2003; Everitt, Markus and Plum, 2005; Grubb, 1968, 1983, 2006].”