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 Ω⊂Rⁿ$\mathrm{\Omega }\subset {\mathbb{R}}^n$, n∈N$n\in \mathbb{N}$, and sufficiently smooth coefficients a,b,q$a,b,q$, we consider the closed, strictly positive, higher-order differential operator 9aaa775cb1ee552fc970147260c5d" title="Click to view the MathML source">A_Ω,2m(a,b,q)$A_{\mathrm{\Omega },2m}(a,b,q)$ in 9aba44d21c895447b153ac28" title="Click to view the MathML source">L²(Ω)$L^2(\mathrm{\Omega })$ defined on 90734e640a190aac9f6c74">$W_0^{2m,2}(\mathrm{\Omega })$, associated with the differential expression and its Krein–von Neumann extension ae4f9caee6dda0d791f" title="Click to view the MathML source">A_K,Ω,2m(a,b,q)$A_{K,\mathrm{\Omega },2m}(a,b,q)$ in 9aba44d21c895447b153ac28" title="Click to view the MathML source">L²(Ω)$L^2(\mathrm{\Omega })$. Denoting by N(λ;A_K,Ω,2m(a,b,q))$N(\lambda ;A_{K,\mathrm{\Omega },2m}(a,b,q))$, e5887571a78720" title="Click to view the MathML source">λ>0$\lambda >0$, the eigenvalue counting function corresponding to the strictly positive eigenvalues of ae4f9caee6dda0d791f" 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 9aad096e5" title="Click to view the MathML source">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 L²(Rⁿ)$L^2({\mathbb{R}}^n)$ defined on W^2m,2(Rⁿ)$W^{2m,2}({\mathbb{R}}^n)$, corresponding to 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 e5ee4c2a0c93f5c7dd08d47" title="Click to view the MathML source">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 9af1b5a5fb">${\tilde{A}}_2(a,b,q)$ in L²(Rⁿ)$L^2({\mathbb{R}}^n)$.

We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension 9aa1004e0773e4317b0b20c7f7e4b3" title="Click to view the MathML source">A_F,Ω,2m(a,b,q)$A_{F,\mathrm{\Omega },2m}(a,b,q)$ in 9aba44d21c895447b153ac28" title="Click to view the MathML source">L²(Ω)$L^2(\mathrm{\Omega })$ of 9aaa775cb1ee552fc970147260c5d" title="Click to view the MathML source">A_Ω,2m(a,b,q)$A_{\mathrm{\Omega },2m}(a,b,q)$.

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