Article
Adv. Math. 223, 1372-1467 (2010)
[DOI: 10.1016/j.aim.2009.10.006]
Spectral Theory for Perturbed Krein Laplacians in Nonsmooth Domains
Mark S. Ashbaugh, Mark S. Ashbaugh, Fritz Gesztesy, Marius Mitrea, Roman Shterenberg, and Gerald Teschl
We study spectral properties for HK,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 C1,r, r>1/2. 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 vn=πn/2/ Γ((n/2)+1) denotes the volume of the unit ball
in ℝn, 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 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 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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