A bound for the eigenvalue counting function for Krein-von Neumann and Friedrichs extensions

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• 作者：Mark S. Ashbaugh^a ; ^{ashbaughm@missouri.edu" class="auth_mail" title="E-mail the corresponding author} ; ; Fritz Gesztesy^a ; ^b ; ^{Fritz_Gesztesy@baylor.edu" class="auth_mail" title="E-mail the corresponding author} ; ; Ari Laptev^c ; ^d ; ¹ ; ^{a.laptev@imperial.ac.uk" class="auth_mail" title="E-mail the corresponding author} ; ; Marius Mitrea^a ; ¹ ; ^{mitream@missouri.edu" class="auth_mail" title="E-mail the corresponding author} ; ; Selim Sukhtaiev^a ; ^{sswfd@mail.missouri.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 35J25 ; 35J40 ; 35P15 ; secondary ; 35P05 ; 46E35 ; 47A10 ; 47F05
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：2 January 2017
• 年：2017
• 卷：304
• 期：Complete
• 页码：1108-1155
• 全文大小：787 K

文摘

For an arbitrary open, nonempty, bounded set 949dd851938f0fc4478d11304a01" title="Click to view the MathML source">Ω⊂Rⁿ$\mathrm{\Omega }\subset {\mathbb{R}}^n$, n∈N$n\in \mathbb{N}$, and sufficiently smooth coefficients e7e787bf1213b235b01262b51" title="Click to view the MathML source">a,b,q$a,b,q$, we consider the closed, strictly positive, higher-order differential operator A_Ω,2m(a,b,q)$A_{\mathrm{\Omega },2m}(a,b,q)$ in a44d21c895447b153ac28" title="Click to view the MathML source">L²(Ω)$L^2(\mathrm{\Omega })$ defined on b42d7a0e6590734e640a190aac9f6c74">$W_0^{2m,2}(\mathrm{\Omega })$, associated with the differential expression and its Krein–von Neumann extension e6dda0d791f" title="Click to view the MathML source">A_K,Ω,2m(a,b,q)$A_{K,\mathrm{\Omega },2m}(a,b,q)$ in a44d21c895447b153ac28" title="Click to view the MathML source">L²(Ω)$L^2(\mathrm{\Omega })$. Denoting by N(λ;A_K,Ω,2m(a,b,q))$N(\lambda ;A_{K,\mathrm{\Omega },2m}(a,b,q))$, λ>0$\lambda >0$, the eigenvalue counting function corresponding to the strictly positive eigenvalues of e6dda0d791f" title="Click to view the MathML source">A_K,Ω,2m(a,b,q)$A_{K,\mathrm{\Omega },2m}(a,b,q)$, we derive the bound where C=C(a,b,q,Ω)>0$C=C(a,b,q,\mathrm{\Omega })>0$ (with C(I_n,0,0,Ω)=|Ω|$C(I_n,0,0,\mathrm{\Omega })=|\mathrm{\Omega }|$) is connected to the eigenfunction expansion of the self-adjoint operator ${\tilde{A}}_{2m}(a,b,q)$ in b045fb410d215a3c8a761dd6" title="Click to view the MathML source">L²(Rⁿ)$L^2({\mathbb{R}}^n)$ defined on a4866bc01139f4d46b" title="Click to view the MathML source">W^2m,2(Rⁿ)$W^{2m,2}({\mathbb{R}}^n)$, corresponding to e600e305f4eb19886d204c2c0a318fe" title="Click to view the MathML source">τ_2m(a,b,q)${\tau }_{2m}(a,b,q)$. Here v_n:=π^n/2/Γ((n+2)/2)$v_n:={\pi }^{n/2}/\mathrm{\Gamma }((n+2)/2)$ denotes the (Euclidean) volume of the unit ball in Rⁿ${\mathbb{R}}^n$.

Our method of proof relies on variational considerations exploiting the fundamental link between the Krein–von Neumann extension and an underlying abstract buckling problem, and on the distorted Fourier transform defined in terms of the eigenfunction transform of ${\tilde{A}}_2(a,b,q)$ in b045fb410d215a3c8a761dd6" title="Click to view the MathML source">L²(Rⁿ)$L^2({\mathbb{R}}^n)$.

We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension b0b20c7f7e4b3" title="Click to view the MathML source">A_F,Ω,2m(a,b,q)$A_{F,\mathrm{\Omega },2m}(a,b,q)$ in a44d21c895447b153ac28" title="Click to view the MathML source">L²(Ω)$L^2(\mathrm{\Omega })$ of A_Ω,2m(a,b,q)$A_{\mathrm{\Omega },2m}(a,b,q)$.

No assumptions on the boundary ∂Ω of Ω are made.