Spectral theory for perturbed Krein Laplacians in nonsmooth domains

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• 作者：Mark S. Ashbaugh ; Fritz Gesztesy ; Marius Mitrea ; Gerald Teschl
• 关键词：Lipschitz domains ; Krein Laplacian ; Eigenvalues ; Spectral analysis ; Weyl asymptotics ; Buckling problem
• 刊名：Advances in Mathematics
• 出版年：2010
• 出版时间：1 March 2010
• 年：2010
• 卷：223
• 期：4
• 页码：1372-1467
• 全文大小：764 K

文摘

We study spectral properties for H_K,Ω, the Krein–von Neumann extension of the perturbed Laplacian −Δ+V defined on , where V is measurable, bounded and nonnegative, in a bounded open set 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 where v_n=π^n/2/Γ((n/2)+1) denotes the volume of the unit ball in , and λ_K,Ω,j, , 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 ) 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 , n3, with compact and vanishing Bessel capacity B_2,2(K)=0, to prove equality of Friedrichs and Krein Laplacians in L²(Ω;dⁿx), that is, has a unique nonnegative self-adjoint extension in L²(Ω;dⁿx).