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 A_Ω,2m(a,b,q)$A_{\mathrm{\Omega },2m}(a,b,q)$ in L²(Ω)$L^2(\mathrm{\Omega })$ defined on $W_0^{2m,2}(\mathrm{\Omega })$, associated with the differential expression and its Krein–von Neumann extension A_K,Ω,2m(a,b,q)$A_{K,\mathrm{\Omega },2m}(a,b,q)$ in 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 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 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 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 L²(Rⁿ)$L^2({\mathbb{R}}^n)$.

We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension A_F,Ω,2m(a,b,q)$A_{F,\mathrm{\Omega },2m}(a,b,q)$ in 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.