Spectral theory for perturbed Krein Laplacians in nonsmooth domains

2009/07/31 by Mark S. Ashbaugh, Fritz Gesztesy, Marius Mitrea +1 · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Ball (mathematics) #Bounded function #Combinatorics #Compact space #Context (archaeology) #Eigenvalues and eigenvectors #Laplace operator #Mathematical analysis #Mathematics #Nonlinear Partial Differential Equations #Pure mathematics #Spectral Theory in Mathematical Physics #Unit sphere #math.AP #math.SP #msc:35J25 #msc:35J40 #msc:35P05 #msc:35P15 #msc:46E35 #msc:47A10 #msc:47F05

paper · pdf · doi:10.1016/j.aim.2009.10.006

published as Adv. Math. 223, 1372-1467 (2010) · 60 pages

openalex publication_date 2009/10/28 · arxiv created 2010/01/25 · arxiv updated 2010/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

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⊂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. 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, 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.

Citations

• Elliptic Partial Differential Equations of Second Order
• Ramifications, old and new, of the eigenvalue problem
• Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices
• Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren
• Krein's resolvent formula for self-adjoint extensions of symmetric second-order elliptic differential operators
• SPECTRA OF SELF-ADJOINT EXTENSIONS AND APPLICATIONS TO SOLVABLE SCHRÖDINGER OPERATORS
• Non-negative perturbations of non-negative self-adjoint operators
• Boundary value problems for elliptic partial differential operators on bounded domains
• Weakly Differentiable Functions

Cited by

• 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
• Some Remarks On Krein--von Neumann Extensions