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 id="mmlsi1" class="mathmlsrc">ixSupport 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⊂Rⁿiner hidden">

img="s

i1.gif" overflow="scroll">i mathvariant="normal">Ωi>⊂i mathvariant="double-struck">Ri>i>ni>, id="mmlsi2" class="mathmlsrc">ixSupport 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∈Niner hidden">

img="s

i2.gif" overflow="scroll">i>ni>∈i mathvariant="double-struck">Ni>, and sufficiently smooth coefficients id="mmlsi3" class="mathmlsrc">ixSupport 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,qiner hidden">

img="s

i3.gif" overflow="scroll">i>ai>,i>bi>,i>qi>, we consider the closed, strictly positive, higher-order differential operator id="mmlsi258" class="mathmlsrc">ixSupport 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)iner hidden">

img="s

i258.gif" overflow="scroll">i>Ai>i mathvariant="normal">Ωi>,2i>mi>(i>ai>,i>bi>,i>qi>) in id="mmlsi369" class="mathmlsrc">ixSupport 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)iner hidden">

img="s

i369.gif" overflow="scroll">i>Li>2(i mathvariant="normal">Ωi>) defined on id="mmlsi6" class="mathmlsrc">itle="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"><img 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">ipt><img 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">ipt>iner hidden">

img="s

i6.gif" overflow="scroll">i>Wi>02i>mi>,2(i mathvariant="normal">Ωi>), associated with the differential expressioniv class="formula" id="fm0010">iv class="mathml">id="mmlsi7" class="mathmlsrc">itle="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"><img 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">ipt><img 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">ipt>iner hidden">

img="s

i7.gif" overflow="scroll">le displaystyle="true" columnspacing="0.2em">ign="left">i>τi>2i>mi>(i>ai>,i>bi>,i>qi>):=ize="5.2ex" minsize="5.2ex">(lelimits="false">∑i>ji>,i>ki>=1i>ni>(−i>ii>∂i>ji>−i>bi>i>ji>)i>ai>i>ji>,i>ki>(−i>ii>∂i>ki>−i>bi>i>ki>)+i>qi>ize="5.2ex" minsize="5.2ex">)i>mi>,ign="left">idth="1em">i>mi>∈i mathvariant="double-struck">Ni>,le><img class="temp" src="/sd/blank.gif">iv>iv> and its Krein–von Neumann extension id="mmlsi11" class="mathmlsrc">ixSupport 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)iner hidden">

img="s

i11.gif" overflow="scroll">i>Ai>i>Ki>,i mathvariant="normal">Ωi>,2i>mi>(i>ai>,i>bi>,i>qi>) in id="mmlsi369" class="mathmlsrc">ixSupport 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)iner hidden">

img="s

i369.gif" overflow="scroll">i>Li>2(i mathvariant="normal">Ωi>). Denoting by id="mmlsi62" class="mathmlsrc">ixSupport 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))iner hidden">

img="s

i62.gif" overflow="scroll">i>Ni>(i>λi>;i>Ai>i>Ki>,i mathvariant="normal">Ωi>,2i>mi>(i>ai>,i>bi>,i>qi>)), id="mmlsi10" class="mathmlsrc">ixSupport 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>0iner hidden">

img="s

i10.gif" overflow="scroll">i>λi>>0, the eigenvalue counting function corresponding to the strictly positive eigenvalues of id="mmlsi11" class="mathmlsrc">ixSupport 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)iner hidden">

img="s

i11.gif" overflow="scroll">i>Ai>i>Ki>,i mathvariant="normal">Ωi>,2i>mi>(i>ai>,i>bi>,i>qi>), we derive the boundiv class="formula" id="fm0020">iv class="mathml">id="mmlsi12" class="mathmlsrc">itle="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"><img 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">ipt><img 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">ipt>iner hidden">

img="s

i12.gif" overflow="scroll">le displaystyle="true" columnspacing="0.1em">ign="right">ign="left">i>Ni>(i>λi>;i>Ai>i>Ki>,i mathvariant="normal">Ωi>,2i>mi>(i>ai>,i>bi>,i>qi>))ign="right">ign="left">idth="1em">≤i>Ci>i>vi>i>ni>(2i>πi>)−i>ni>ize="5.2ex" minsize="5.2ex">(1+2i>mi>2i>mi>+i>ni>ize="5.2ex" minsize="5.2ex">)i>ni>/(2i>mi>)i>λi>i>ni>/(2i>mi>),ign="right">ign="left">idth="2em">i>λi>>0,le><img class="temp" src="/sd/blank.gif">iv>iv> where id="mmlsi13" class="mathmlsrc">ixSupport 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)>0iner hidden">

img="s

i13.gif" overflow="scroll">i>Ci>=i>Ci>(i>ai>,i>bi>,i>qi>,i mathvariant="normal">Ωi>)>0 (with id="mmlsi14" class="mathmlsrc">ixSupport 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|iner hidden">

img="s

i14.gif" overflow="scroll">i>Ci>(i>Ii>i>ni>,0,0,i mathvariant="normal">Ωi>)=|i mathvariant="normal">Ωi>|) is connected to the eigenfunction expansion of the self-adjoint operator id="mmlsi15" class="mathmlsrc">itle="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"><img 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">ipt><img 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">ipt>iner hidden">

img="s

i15.gif" overflow="scroll">i>Ai>&tilde;2i>mi>(i>ai>,i>bi>,i>qi>) in id="mmlsi16" class="mathmlsrc">ixSupport 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)iner hidden">

img="s

i16.gif" overflow="scroll">i>Li>2(i mathvariant="double-struck">Ri>i>ni>) defined on id="mmlsi17" class="mathmlsrc">ixSupport 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)iner hidden">

img="s

i17.gif" overflow="scroll">i>Wi>2i>mi>,2(i mathvariant="double-struck">Ri>i>ni>), corresponding to id="mmlsi18" class="mathmlsrc">ixSupport 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)iner hidden">

img="s

i18.gif" overflow="scroll">i>τi>2i>mi>(i>ai>,i>bi>,i>qi>). Here id="mmlsi19" class="mathmlsrc">ixSupport 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)iner hidden">

img="s

i19.gif" overflow="scroll">i>vi>i>ni>:=i>πi>i>ni>/2/i mathvariant="normal">Γi>((i>ni>+2)/2) denotes the (Euclidean) volume of the unit ball in id="mmlsi20" class="mathmlsrc">ixSupport 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">Rⁿiner hidden">

img="s

i20.gif" overflow="scroll">i mathvariant="double-struck">Ri>i>ni>.

id="sp0020">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 id="mmlsi21" class="mathmlsrc">itle="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"><img 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">ipt><img height="20" border="0" style="vertical-align:bottom" width="70" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0001870816305412-si21.gif">ipt>iner hidden"> $img="s$ i21.gif" overflow="scroll">i>Ai>&tilde;2(i>ai>,i>bi>,i>qi>) in id="mmlsi16" class="mathmlsrc">ixSupport 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)iner hidden"> $img="s$ i16.gif" overflow="scroll">i>Li>2(i mathvariant="double-struck">Ri>i>ni>).

id="sp0030">We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension id="mmlsi22" class="mathmlsrc">ixSupport 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)iner hidden"> $img="s$ i22.gif" overflow="scroll">i>Ai>i>Fi>,i mathvariant="normal">Ωi>,2i>mi>(i>ai>,i>bi>,i>qi>) in id="mmlsi369" class="mathmlsrc">ixSupport 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)iner hidden"> $img="s$ i369.gif" overflow="scroll">i>Li>2(i mathvariant="normal">Ωi>) of id="mmlsi258" class="mathmlsrc">ixSupport 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)iner hidden"> $img="s$ i258.gif" overflow="scroll">i>Ai>i mathvariant="normal">Ωi>,2i>mi>(i>ai>,i>bi>,i>qi>).

id="sp0040">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