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 ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si1.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=6b9e949dd851938f0fc4478d11304a01" title="Click to view the MathML source">Ω⊂Rⁿ$\mathrm{\Omega }\subset {\mathbb{R}}^n$, ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si2.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=69c9ef5221267c27b13c5eb2e8213826" title="Click to view the MathML source">n∈N$n\in \mathbb{N}$, and sufficiently smooth coefficients ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si3.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=1cb6493e7e787bf1213b235b01262b51" title="Click to view the MathML source">a,b,q$a,b,q$, we consider the closed, strictly positive, higher-order differential operator ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si258.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=0929aaa775cb1ee552fc970147260c5d" title="Click to view the MathML source">A_Ω,2m(a,b,q)$A_{\mathrm{\Omega },2m}(a,b,q)$ in ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si369.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=d121ec289aba44d21c895447b153ac28" title="Click to view the MathML source">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 ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si11.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=e2b56797399b5ae4f9caee6dda0d791f" title="Click to view the MathML source">A_K,Ω,2m(a,b,q)$A_{K,\mathrm{\Omega },2m}(a,b,q)$ in ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si369.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=d121ec289aba44d21c895447b153ac28" title="Click to view the MathML source">L²(Ω)$L^2(\mathrm{\Omega })$. Denoting by ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si62.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=cdc449ee02c64da54ea89b5ac8934320" title="Click to view the MathML source">N(λ;A_K,Ω,2m(a,b,q))$N(\lambda ;A_{K,\mathrm{\Omega },2m}(a,b,q))$, ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si10.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=54b878bd15ec619da6e5887571a78720" title="Click to view the MathML source">λ>0$\lambda >0$, the eigenvalue counting function corresponding to the strictly positive eigenvalues of ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si11.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=e2b56797399b5ae4f9caee6dda0d791f" 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 ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si13.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=06a37e118520b7bf93583079aad096e5" title="Click to view the MathML source">C=C(a,b,q,Ω)>0$C=C(a,b,q,\mathrm{\Omega })>0$ (with ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si14.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=69e16cf5a79976c0bdc5bc08b8e48a3a" title="Click to view the MathML source">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 22431b52517af80">${\tilde{A}}_{2m}(a,b,q)$ in ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si16.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=5bd6d975b045fb410d215a3c8a761dd6" title="Click to view the MathML source">L²(Rⁿ)$L^2({\mathbb{R}}^n)$ defined on ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si17.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=f8d3fa08ca5c2aa4866bc01139f4d46b" title="Click to view the MathML source">W^2m,2(Rⁿ)$W^{2m,2}({\mathbb{R}}^n)$, corresponding to ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si18.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=2e600e305f4eb19886d204c2c0a318fe" title="Click to view the MathML source">τ_2m(a,b,q)${\tau }_{2m}(a,b,q)$. Here ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si19.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=a1a8e064040393224564798b3223ed71" title="Click to view the MathML source">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 ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si20.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=c87582d62e5ee4c2a0c93f5c7dd08d47" 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 ${\tilde{A}}_2(a,b,q)$ in ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si16.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=5bd6d975b045fb410d215a3c8a761dd6" 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 22" class="mathmlsrc">ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si22.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=509aa1004e0773e4317b0b20c7f7e4b3" title="Click to view the MathML source">A_F,Ω,2m(a,b,q) in ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si369.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=d121ec289aba44d21c895447b153ac28" title="Click to view the MathML source">L²(Ω)$L^2(\mathrm{\Omega })$ of ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si258.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=0929aaa775cb1ee552fc970147260c5d" 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.