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

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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 class="mathmlsrc">class="formulatext 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⊂Rⁿclass="mathContainer hidden">class="mathCode">

verflow="scroll"> variant="normal">Ω \subset {variant="double-struck">R}^{n}

, class="mathmlsrc">class="formulatext 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∈Nclass="mathContainer hidden">class="mathCode">

verflow="scroll"> n \in variant="double-struck">N

, and sufficiently smooth coefficients class="mathmlsrc">class="formulatext 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,qclass="mathContainer hidden">class="mathCode">

verflow="scroll"> a, b, q

, we consider the closed, strictly positive, higher-order differential operator class="mathmlsrc">class="formulatext 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)class="mathContainer hidden">class="mathCode">

verflow="scroll"> A_{variant="normal">Ω, 2 m} (a, b, q)

in class="mathmlsrc">class="formulatext 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)class="mathContainer hidden">class="mathCode">

verflow="scroll"> L^{2} (variant="normal">Ω)

defined on class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si6.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=b42d7a0e6590734e640a190aac9f6c74">class="imgLazyJSB inlineImage" height="20" width="71" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816305412-si6.gif">

thetitlethe

class="mathContainer hidden">class="mathCode">

verflow="scroll"> W_{0}^{2 m, 2} (variant="normal">Ω)

, associated with the differential expressionv class="formula" id="fm0010">v class="mathml">class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si7.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=f99eca460a35ef3d07681c7f758cce72">class="imgLazyJSB inlineImage" height="72" width="393" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816305412-si7.gif">

thetitlethe

class="mathContainer hidden">class="mathCode">

verflow="scroll"> \begin{matrix} τ_{2 m} (a, b, q) : = {(ver> vablelimits="false">\sum j, k = 1 n ver> (- i \partial_{j} - b_{j}) a_{j, k} (- i \partial_{k} - b_{k}) + q)}^{m}, \\ m \in variant="double-struck">N, \end{matrix}

class="temp" src="/sd/blank.gif">v>v> and its Krein–von Neumann extension class="mathmlsrc">class="formulatext 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)class="mathContainer hidden">class="mathCode">

verflow="scroll"> A_{K, variant="normal">Ω, 2 m} (a, b, q)

verflow="scroll"> L^{2} (variant="normal">Ω)

. Denoting by class="mathmlsrc">class="formulatext 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))class="mathContainer hidden">class="mathCode">

verflow="scroll"> N (λ; A_{K, variant="normal">Ω, 2 m} (a, b, q))

, class="mathmlsrc">class="formulatext 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>0class="mathContainer hidden">class="mathCode">

verflow="scroll"> λ > 0

, the eigenvalue counting function corresponding to the strictly positive eigenvalues of class="mathmlsrc">class="formulatext 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)class="mathContainer hidden">class="mathCode">

verflow="scroll"> A_{K, variant="normal">Ω, 2 m} (a, b, q)

, we derive the boundv class="formula" id="fm0020">v class="mathml">class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si12.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=6a52b64ff7acd6ed5dbad744148490d4">class="imgLazyJSB inlineImage" height="93" width="325" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816305412-si12.gif">

thetitlethe

class="mathContainer hidden">class="mathCode">

verflow="scroll"> \begin{matrix} N (λ; A_{K, variant="normal">Ω, 2 m} (a, b, q)) \\ \leq C v_{n} {(2 π)}^{- n} {(1 + \frac{2 m}{2 m + n})}^{n / (2 m)} λ^{n / (2 m)}, \\ λ > 0, \end{matrix}

class="temp" src="/sd/blank.gif">v>v> where class="mathmlsrc">class="formulatext 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)>0class="mathContainer hidden">class="mathCode">

verflow="scroll"> C = C (a, b, q, variant="normal">Ω) > 0

(with class="mathmlsrc">class="formulatext 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|class="mathContainer hidden">class="mathCode">

verflow="scroll"> C (I_{n}, 0, 0, variant="normal">Ω) = | variant="normal">Ω |

) is connected to the eigenfunction expansion of the self-adjoint operator class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si15.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=1c56f34c60fab11ca22431b52517af80">class="imgLazyJSB inlineImage" height="20" width="82" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816305412-si15.gif">

thetitlethe

class="mathContainer hidden">class="mathCode">

verflow="scroll"> {ver accent="true"> A ˜ ver>}_{2 m} (a, b, q)

in class="mathmlsrc">class="formulatext 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)class="mathContainer hidden">class="mathCode">

verflow="scroll"> L^{2} ({variant="double-struck">R}^{n})

defined on class="mathmlsrc">class="formulatext 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)class="mathContainer hidden">class="mathCode">

verflow="scroll"> W^{2 m, 2} ({variant="double-struck">R}^{n})

, corresponding to class="mathmlsrc">class="formulatext 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)class="mathContainer hidden">class="mathCode">

verflow="scroll"> τ_{2 m} (a, b, q)

. Here class="mathmlsrc">class="formulatext 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)class="mathContainer hidden">class="mathCode">

verflow="scroll"> v_{n} : = π^{n / 2} / variant="normal">Γ ((n + 2) / 2)

denotes the (Euclidean) volume of the unit ball in class="mathmlsrc">class="formulatext 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">Rⁿclass="mathContainer hidden">class="mathCode">

verflow="scroll"> {variant="double-struck">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 class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305412&_mathId=si21.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=1762617bddf196643438789af1b5a5fb">class="imgLazyJSB inlineImage" height="20" width="70" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816305412-si21.gif">class="mathContainer hidden">class="mathCode"> $verflow="scroll"> {ver accent="true"> A ˜ ver>}_{2} (a, b, q)$ in class="mathmlsrc">class="formulatext 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)class="mathContainer hidden">class="mathCode"> $verflow="scroll"> L^{2} ({variant="double-struck">R}^{n})$ .

We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension class="mathmlsrc">class="formulatext 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)class="mathContainer hidden">class="mathCode"> $verflow="scroll"> A_{F, variant="normal">Ω, 2 m} (a, b, q)$ in class="mathmlsrc">class="formulatext 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)class="mathContainer hidden">class="mathCode"> $verflow="scroll"> L^{2} (variant="normal">Ω)$ of class="mathmlsrc">class="formulatext 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)class="mathContainer hidden">class="mathCode"> $verflow="scroll"> A_{variant="normal">Ω, 2 m} (a, b, q)$ .

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