锥中与稳态的薛定谔算子相关的广义Martin函数无穷远处的控制（英文）
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摘要
本文研究了稳态的薛定谔算子的Dirichlet问题和Martin函数的边界行为.利用广义Martin表示和稳态的薛定谔算子对应的常微分方程基本解,在具有光滑边界的锥形区域中获得了与稳态的薛定谔算子相关的广义Martin函数无穷远处广义调和控制的一些刻画,推广了拉普拉斯算子情形的结果.
In the paper, we mainly study Dirichlet problem for the stationary Schrdinger operator and the boundary behavior of Martin function. Depended on the generalized Martin representation and the fundamental system of solutions of an ordinary differential equation corresponding to stationary Schrdinger operator, we obtain some characterizations for the majorization of the generalized Martin functions associated with the stationary Schrdinger operator in a cone with smooth boundary, and generalize some classical results in Laplace setting.
In the paper, we mainly study Dirichlet problem for the stationary Schrdinger operator and the boundary behavior of Martin function. Depended on the generalized Martin representation and the fundamental system of solutions of an ordinary differential equation corresponding to stationary Schrdinger operator, we obtain some characterizations for the majorization of the generalized Martin functions associated with the stationary Schrdinger operator in a cone with smooth boundary, and generalize some classical results in Laplace setting.
引文
[1]Azarin V S.Generalization of a theorem of Hayman on subharmonic functions in an m-dimensional cone[J].Trans.Amer.Math.Soc.,1969,80(2):119-138.
[2]Aikawa H,Essén M.Potential theory-selected topics[M].Lect.Notes Math.,1633,Berlin:SpringerVerlag,1996.
[3]Armitage D H,Gardiner S J.Classical potential theory[M].London:Springer-verlag,2001.
[4]Armitage D H,Kuran.On positive harmonic majorization of y in Rn×(0,∞)[J].J.London Math.Soc.Ser.II,1971,3:733-741.
[5]Dalberg B E J.A minimum principle for positive harmonic functions[J].Proc.London Math.Soc.,1976,33(2):238-250.
[6]Gilbarg D,Trudinger N S.Elliptic partial differential equations of second order[M].Berlin:Springer Verlag,1977.
[7]Long Pinhong,Gao Zhiqiang,Deng Guantie.Criteria of Wiener type for minimally thin sets and rarefied sets associated with the stationary Schrdinger operator in a cone[J].Abstr.Appl.Anal.,DOI:10.1155/2012/453891.
[8]Long Pinhong.Characterizations for exceptional sets and growth problems in classical or nonlinear potential theory[D].Beijng:Beijing Normal University,2013.
[9]Levin B,Kheyfits A.Asymptotic behavior of subfunctions of the stationary Schrdinger operator[J].http://arxiv.org/abs/math/0211328v1,2002,96 pp.
[10]Miyamoto I,Yanagishita M,Yoshida H.On harmonic majorization of the Martin function at infnity in a cone[J].Czech.Math.J.,2005,55(130):1041-1054.
[11]Miyamoto I,Yoshida H.Two criterions of Wiener type for minimally thin sets and rarefied sets in a cone[J].J.Math.Soc.Japan.,2002,54(3):487-512.
[12]Miyamoto I,Yoshida H.On a-minimally thin sets at infinity in a cone[J].Hiroshima Math.J.,2007,37(1):61-80.
[13]Qiao Lei.Some researches on(generalized)harmonic and superharmonic functions[D].Beijng:Beijing Normal University,2010.
[14]Qiao Lei,Deng Guantie.A theorem of Phragmén Lindelf type for subfunctions in a cone[J].Glasg.Math.J.,2011,53(3):599-610.
[15]Qiao Lei,Deng Guantie.The Riesz decomposition theorem for superharmonic functions in a cone and its application(in Chinese)[J].Scientia Sinica(Math.),2012,42(8):763-774.
[16]Qiao Lei,Deng Guantie.Integral representation for the solution of the stationary Schrodinger equation in a cone[J].Math.Nachr.,2012,285(16):2029-2038.
[17]Qiao Lei,Deng Guan Tie,Pan Guo Shuang.Exceptional sets of modified Poisson integral and Green potential in the upper-half space[J].Scientia Sinica(Math.),2010,40(8):787-800.
[18]Reed M,Simon B.Methods of modern mathematical physics[M].Vol.3,London,New York,San Francisco:Acad.Press,1970.
[19]Rosenblum G V,Solomyak M Z,Shubin M A.Spectral theory of differential operators[M].Moscow:VINITI,1989.
[20]Simon B,Schrdinger semigroups[J].Bull.Amer.Math.Soc.,1982,7:447-526.
[21]Stein E M.Singular integrals and differentiability properties of functions[M].New Jersey:Princeton University Press,1970.
[22]Verzhbinskii G M,Maz’ya V G.Asymptotic behavior of solutions of elliptic equations of the second order close to a boundary.I[J].Sibirsk.Matem.Zh.,1971,12(6):1217-1249.
[23]Zhang Yanhui,Deng Guantie.Growth properties of generalized Poisson integral in the half space[J].J.Math.,2013,33(3):473-478.
[2]Aikawa H,Essén M.Potential theory-selected topics[M].Lect.Notes Math.,1633,Berlin:SpringerVerlag,1996.
[3]Armitage D H,Gardiner S J.Classical potential theory[M].London:Springer-verlag,2001.
[4]Armitage D H,Kuran.On positive harmonic majorization of y in Rn×(0,∞)[J].J.London Math.Soc.Ser.II,1971,3:733-741.
[5]Dalberg B E J.A minimum principle for positive harmonic functions[J].Proc.London Math.Soc.,1976,33(2):238-250.
[6]Gilbarg D,Trudinger N S.Elliptic partial differential equations of second order[M].Berlin:Springer Verlag,1977.
[7]Long Pinhong,Gao Zhiqiang,Deng Guantie.Criteria of Wiener type for minimally thin sets and rarefied sets associated with the stationary Schrdinger operator in a cone[J].Abstr.Appl.Anal.,DOI:10.1155/2012/453891.
[8]Long Pinhong.Characterizations for exceptional sets and growth problems in classical or nonlinear potential theory[D].Beijng:Beijing Normal University,2013.
[9]Levin B,Kheyfits A.Asymptotic behavior of subfunctions of the stationary Schrdinger operator[J].http://arxiv.org/abs/math/0211328v1,2002,96 pp.
[10]Miyamoto I,Yanagishita M,Yoshida H.On harmonic majorization of the Martin function at infnity in a cone[J].Czech.Math.J.,2005,55(130):1041-1054.
[11]Miyamoto I,Yoshida H.Two criterions of Wiener type for minimally thin sets and rarefied sets in a cone[J].J.Math.Soc.Japan.,2002,54(3):487-512.
[12]Miyamoto I,Yoshida H.On a-minimally thin sets at infinity in a cone[J].Hiroshima Math.J.,2007,37(1):61-80.
[13]Qiao Lei.Some researches on(generalized)harmonic and superharmonic functions[D].Beijng:Beijing Normal University,2010.
[14]Qiao Lei,Deng Guantie.A theorem of Phragmén Lindelf type for subfunctions in a cone[J].Glasg.Math.J.,2011,53(3):599-610.
[15]Qiao Lei,Deng Guantie.The Riesz decomposition theorem for superharmonic functions in a cone and its application(in Chinese)[J].Scientia Sinica(Math.),2012,42(8):763-774.
[16]Qiao Lei,Deng Guantie.Integral representation for the solution of the stationary Schrodinger equation in a cone[J].Math.Nachr.,2012,285(16):2029-2038.
[17]Qiao Lei,Deng Guan Tie,Pan Guo Shuang.Exceptional sets of modified Poisson integral and Green potential in the upper-half space[J].Scientia Sinica(Math.),2010,40(8):787-800.
[18]Reed M,Simon B.Methods of modern mathematical physics[M].Vol.3,London,New York,San Francisco:Acad.Press,1970.
[19]Rosenblum G V,Solomyak M Z,Shubin M A.Spectral theory of differential operators[M].Moscow:VINITI,1989.
[20]Simon B,Schrdinger semigroups[J].Bull.Amer.Math.Soc.,1982,7:447-526.
[21]Stein E M.Singular integrals and differentiability properties of functions[M].New Jersey:Princeton University Press,1970.
[22]Verzhbinskii G M,Maz’ya V G.Asymptotic behavior of solutions of elliptic equations of the second order close to a boundary.I[J].Sibirsk.Matem.Zh.,1971,12(6):1217-1249.
[23]Zhang Yanhui,Deng Guantie.Growth properties of generalized Poisson integral in the half space[J].J.Math.,2013,33(3):473-478.