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

详细信息查看全文

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 g" 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

g="si1.

gif" overflow="scroll">Ω⊂Rn, g" 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

g="si2.

gif" overflow="scroll">n∈N, and sufficiently smooth coefficients g" 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

g="si3.

gif" overflow="scroll">a,b,q, we consider the closed, strictly positive, higher-order differential operator g" 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)

g="si258.

gif" overflow="scroll">AΩ,2m(a,b,q) in g" 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)

g="si369.

gif" overflow="scroll">L2(Ω) defined on g" 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">g 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">

g height="20" border="0" style="vertical-align:bottom" width="71" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0001870816305412-si6.gif">

g="si6.

gif" overflow="scroll">W02m,2(Ω), associated with the differential expression

g" 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">g 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">

g height="72" border="0" style="vertical-align:bottom" width="393" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0001870816305412-si7.gif">

g="si7.

gif" overflow="scroll">g="0.2em">gn="left">τ2m(a,b,q):=(∑j,k=1n(−i∂j−bj)aj,k(−i∂k−bk)+q)m,gn="left">m∈N,g class="temp" src="/sd/blank.gif">

and its Krein–von Neumann extension g" 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)

g="si11.

gif" overflow="scroll">AK,Ω,2m(a,b,q) in g" 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)

g="si369.

gif" overflow="scroll">L2(Ω). Denoting by g" 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))

g="si62.

gif" overflow="scroll">N(λ;AK,Ω,2m(a,b,q)), g" 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

g="si10.

gif" overflow="scroll">λ>0, the eigenvalue counting function corresponding to the strictly positive eigenvalues of g" 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)

g="si11.

gif" overflow="scroll">AK,Ω,2m(a,b,q), we derive the bound

g" 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">g 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">

g height="93" border="0" style="vertical-align:bottom" width="325" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0001870816305412-si12.gif">

g="si12.

gif" overflow="scroll">g="0.1em">gn="right">gn="left">N(λ;AK,Ω,2m(a,b,q))gn="right">gn="left">≤Cvn(2π)−n(1+2m2m+n)n/(2m)λn/(2m),gn="right">gn="left">λ>0,g class="temp" src="/sd/blank.gif">

where g" 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

g="si13.

gif" overflow="scroll">C=C(a,b,q,Ω)>0 (with g" 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|

g="si14.

gif" overflow="scroll">C(In,0,0,Ω)=|Ω|) is connected to the eigenfunction expansion of the self-adjoint operator g" 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">g 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">

g height="20" border="0" style="vertical-align:bottom" width="82" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0001870816305412-si15.gif">

g="si15.

gif" overflow="scroll">A˜2m(a,b,q) in g" 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)

g="si16.

gif" overflow="scroll">L2(Rn) defined on g" 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)

g="si17.

gif" overflow="scroll">W2m,2(Rn), corresponding to g" 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)

g="si18.

gif" overflow="scroll">τ2m(a,b,q). Here g" 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)

g="si19.

gif" overflow="scroll">vn:=πn/2/Γ((n+2)/2) denotes the (Euclidean) volume of the unit ball in g" 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

g="si20.

gif" overflow="scroll">Rn.

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 g" 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">g 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"> $g="si21.$ gif" overflow="scroll">A˜2(a,b,q) in g" 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) $g="si16.$ gif" overflow="scroll">L2(Rn).

We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension g" 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) $g="si22.$ gif" overflow="scroll">AF,Ω,2m(a,b,q) in g" 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) $g="si369.$ gif" overflow="scroll">L2(Ω) of g" 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) $g="si258.$ gif" overflow="scroll">AΩ,2m(a,b,q).

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

地址：北京市海淀区学院路29号邮编：100083

电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700