• Previous Article
    On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ
  • DCDS-B Home
  • This Issue
  • Next Article
    Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity
August  2017, 22(6): 2321-2338. doi: 10.3934/dcdsb.2017099

Asymptotic behaviors of Green-Sch potentials at infinity and its applications

School of of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou 450046, China

Received  June 2016 Revised  November 2016 Published  March 2017

Fund Project: The author is supported by the National Natural Science Foundation of China (Grant Nos. 11301140, U1304102)

The first aim in this paper is to deal with asymptotic behaviors of Green-Sch potentials in a cylinder. As an application we prove the integral representation of nonnegative weak solutions of the stationary Schrödinger equation in a cylinder. Next we give asymptotic behaviors of them outside an exceptional set. Finally we obtain a quantitative property of rarefied sets with respect to the stationary Schrödinger operator at $+\infty$ in a cylinder. Meanwhile we show that the reverse of this property is not true.

Citation: Lei Qiao. Asymptotic behaviors of Green-Sch potentials at infinity and its applications. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2321-2338. doi: 10.3934/dcdsb.2017099
References:
[1]

L. Ahlfors and M. Heins, Questions of regularity connected with the Phragmén-Lindelöf principle, Ann. of Math., 50 (1949), 341-346. Google Scholar

[2]

H. Aikawa, On the behavior at infinity of nonnegative superharmonic functions in a half space, Hiroshima Math. J., 11 (1981), 425-441. Google Scholar

[3]

H. Aikawa and M. Essén, Potential theory-selected topics. Lecture Notes in Mathematics, 1633, Springer-Verlag, Berlin, 1996.Google Scholar

[4]

V. S. Azarin, Generalization of a theorem of Hayman's on a subharmonic function in an n-dimensional cone (Russian), Mat. Sb. (N.S.), 66 (1965), 248-264. Google Scholar

[5]

R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. I. Interscience Publishers, Inc. , New York, N. Y. , 1953.Google Scholar

[6]

M. Cranston, Conditional Brownian motion, Whitney squares and the conditional gauge theorem, Seminar on Stochastic Processes, 1988 (Gainesville, FL, 1988), 109-119, Progr. Probab. , 17, Birkhäuser Boston, Boston, MA, 1989.Google Scholar

[7]

M. CranstonE. Fabes and Z. Zhao, Conditional gauge and potential theory for the Schrödinger operator, Trans. Amer. Math. Soc., 307 (1988), 171-194. Google Scholar

[8]

M. Essén and H. L. Jackson, On the covering properties of certain exceptional sets in a half-space, Hiroshima Math. J., 10 (1980), 233-262. Google Scholar

[9]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.Google Scholar

[10]

P. Hartman, Ordinary Differential Equations, John Wiley and Sons, Inc. , New York-LondonSydney, 1964.Google Scholar

[11]

J. Lelong-Ferrand, Étude au voisinage de la frontière des fonctions surharmoniques positives dans un demi-espace, Ann. Sci. École Norm. Sup., 66 (1949), 125-159. Google Scholar

[12]

B. Ya. Levin and A. I. Kheyfits, Asymptotic behavior of subfunctions of time-independent Schrödinger operator, in Some Topics on Value Distribution and Differentiability in Complex and P-adic Analysis (eds. Escassut A., Tutschke W., C. C. Yang), Science Press, 11 (2008), 323-397. Google Scholar

[13]

I. Miyamoto and H. Yoshida, Two criterions of Wiener type for minimally thin sets and rarefied sets in a cone, J. Math. Soc. Japan., 54 (2002), 487-512. Google Scholar

[14]

I. Miyamoto, Two criteria of Wiener type for minimally thin sets and rarefied sets in a cylinder, Hokkaido Math. J., 36 (2007), 507-534. Google Scholar

[15]

Y. Mizuta, Potential theory in Euclidean spaces. GAKUTO International Series. Mathematical Sciences and Applications, 6, Gakkötosho Co. , Ltd. , Tokyo, 1996.Google Scholar

[16]

L. Qiao, Weak solutions for the stationary Schrödinger equation and its application, Appl. Math. Lett., 63 (2017), 34-39. Google Scholar

[17]

L. Qiao and G. Deng, Growth properties of modified α-potentials in the upper-half space, Filomat, 27 (2013), 703-712. Google Scholar

[18]

L. Qiao and G. Deng, Minimally thin sets at infinity with respect to the Schrödinger operator, Sci. Sin. Math., 44 (2014), 1247-1256. Google Scholar

[19]

L. Qiao and G. Pan, Integral representations of generalized harmonic functions, Taiwanese J. Math., 17 (2013), 1503-1521. Google Scholar

[20]

L. Qiao and G. Pan, Lower-bound estimates for a class of harmonic functions and applications to Masaev's type theorem, Bull. Sci. Math., 140 (2016), 70-85. Google Scholar

[21]

L. Qiao and Y. Ren, ntegral representations for the solutions of infinite order of the stationary Schrödinger equation in a cone, Monats. Math., 173 (2014), 593-603. Google Scholar

[22]

B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc., 7 (1982), 447-526. Google Scholar

[23]

H. Yoshida and I. Miyamoto, Solutions of the Dirichlet problem on a cone with continuous data, J. Math. Soc. Japan, 50 (1998), 71-93. Google Scholar

[24]

Y. ZhangG. Deng and K. Kou, Asymptotic behavior of fractional Laplacians in the half space, Appl. Math. Comput., 254 (2015), 125-132. Google Scholar

[25]

Y. ZhangG. Deng and T. Qian, Integral representations of a class of harmonic functions in the half space, J. Differential Equations, 260 (2016), 923-936. Google Scholar

show all references

References:
[1]

L. Ahlfors and M. Heins, Questions of regularity connected with the Phragmén-Lindelöf principle, Ann. of Math., 50 (1949), 341-346. Google Scholar

[2]

H. Aikawa, On the behavior at infinity of nonnegative superharmonic functions in a half space, Hiroshima Math. J., 11 (1981), 425-441. Google Scholar

[3]

H. Aikawa and M. Essén, Potential theory-selected topics. Lecture Notes in Mathematics, 1633, Springer-Verlag, Berlin, 1996.Google Scholar

[4]

V. S. Azarin, Generalization of a theorem of Hayman's on a subharmonic function in an n-dimensional cone (Russian), Mat. Sb. (N.S.), 66 (1965), 248-264. Google Scholar

[5]

R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. I. Interscience Publishers, Inc. , New York, N. Y. , 1953.Google Scholar

[6]

M. Cranston, Conditional Brownian motion, Whitney squares and the conditional gauge theorem, Seminar on Stochastic Processes, 1988 (Gainesville, FL, 1988), 109-119, Progr. Probab. , 17, Birkhäuser Boston, Boston, MA, 1989.Google Scholar

[7]

M. CranstonE. Fabes and Z. Zhao, Conditional gauge and potential theory for the Schrödinger operator, Trans. Amer. Math. Soc., 307 (1988), 171-194. Google Scholar

[8]

M. Essén and H. L. Jackson, On the covering properties of certain exceptional sets in a half-space, Hiroshima Math. J., 10 (1980), 233-262. Google Scholar

[9]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.Google Scholar

[10]

P. Hartman, Ordinary Differential Equations, John Wiley and Sons, Inc. , New York-LondonSydney, 1964.Google Scholar

[11]

J. Lelong-Ferrand, Étude au voisinage de la frontière des fonctions surharmoniques positives dans un demi-espace, Ann. Sci. École Norm. Sup., 66 (1949), 125-159. Google Scholar

[12]

B. Ya. Levin and A. I. Kheyfits, Asymptotic behavior of subfunctions of time-independent Schrödinger operator, in Some Topics on Value Distribution and Differentiability in Complex and P-adic Analysis (eds. Escassut A., Tutschke W., C. C. Yang), Science Press, 11 (2008), 323-397. Google Scholar

[13]

I. Miyamoto and H. Yoshida, Two criterions of Wiener type for minimally thin sets and rarefied sets in a cone, J. Math. Soc. Japan., 54 (2002), 487-512. Google Scholar

[14]

I. Miyamoto, Two criteria of Wiener type for minimally thin sets and rarefied sets in a cylinder, Hokkaido Math. J., 36 (2007), 507-534. Google Scholar

[15]

Y. Mizuta, Potential theory in Euclidean spaces. GAKUTO International Series. Mathematical Sciences and Applications, 6, Gakkötosho Co. , Ltd. , Tokyo, 1996.Google Scholar

[16]

L. Qiao, Weak solutions for the stationary Schrödinger equation and its application, Appl. Math. Lett., 63 (2017), 34-39. Google Scholar

[17]

L. Qiao and G. Deng, Growth properties of modified α-potentials in the upper-half space, Filomat, 27 (2013), 703-712. Google Scholar

[18]

L. Qiao and G. Deng, Minimally thin sets at infinity with respect to the Schrödinger operator, Sci. Sin. Math., 44 (2014), 1247-1256. Google Scholar

[19]

L. Qiao and G. Pan, Integral representations of generalized harmonic functions, Taiwanese J. Math., 17 (2013), 1503-1521. Google Scholar

[20]

L. Qiao and G. Pan, Lower-bound estimates for a class of harmonic functions and applications to Masaev's type theorem, Bull. Sci. Math., 140 (2016), 70-85. Google Scholar

[21]

L. Qiao and Y. Ren, ntegral representations for the solutions of infinite order of the stationary Schrödinger equation in a cone, Monats. Math., 173 (2014), 593-603. Google Scholar

[22]

B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc., 7 (1982), 447-526. Google Scholar

[23]

H. Yoshida and I. Miyamoto, Solutions of the Dirichlet problem on a cone with continuous data, J. Math. Soc. Japan, 50 (1998), 71-93. Google Scholar

[24]

Y. ZhangG. Deng and K. Kou, Asymptotic behavior of fractional Laplacians in the half space, Appl. Math. Comput., 254 (2015), 125-132. Google Scholar

[25]

Y. ZhangG. Deng and T. Qian, Integral representations of a class of harmonic functions in the half space, J. Differential Equations, 260 (2016), 923-936. Google Scholar

[1]

Xiaoyan Lin, Yubo He, Xianhua Tang. Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1547-1565. doi: 10.3934/cpaa.2019074

[2]

Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377

[3]

Ahmed Y. Abdallah. Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems. Communications on Pure & Applied Analysis, 2006, 5 (1) : 55-69. doi: 10.3934/cpaa.2006.5.55

[4]

Nakao Hayashi, Pavel I. Naumkin. Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 383-400. doi: 10.3934/dcds.1997.3.383

[5]

Haoyue Cui, Dongyi Liu, Genqi Xu. Asymptotic behavior of a Schrödinger equation under a constrained boundary feedback. Mathematical Control & Related Fields, 2018, 8 (2) : 383-395. doi: 10.3934/mcrf.2018015

[6]

Helge Krüger. Asymptotic of gaps at small coupling and applications of the skew-shift Schrödinger operator. Conference Publications, 2011, 2011 (Special) : 874-880. doi: 10.3934/proc.2011.2011.874

[7]

Zhitao Zhang, Haijun Luo. Symmetry and asymptotic behavior of ground state solutions for schrödinger systems with linear interaction. Communications on Pure & Applied Analysis, 2018, 17 (3) : 787-806. doi: 10.3934/cpaa.2018040

[8]

Jean-Claude Saut, Jun-Ichi Segata. Asymptotic behavior in time of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 219-239. doi: 10.3934/dcds.2019009

[9]

Nguyen Dinh Cong, Roberta Fabbri. On the spectrum of the one-dimensional Schrödinger operator. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 541-554. doi: 10.3934/dcdsb.2008.9.541

[10]

Grégoire Allaire, M. Vanninathan. Homogenization of the Schrödinger equation with a time oscillating potential. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 1-16. doi: 10.3934/dcdsb.2006.6.1

[11]

Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003

[12]

Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063

[13]

Marco A. S. Souto, Sérgio H. M. Soares. Ground state solutions for quasilinear stationary Schrödinger equations with critical growth. Communications on Pure & Applied Analysis, 2013, 12 (1) : 99-116. doi: 10.3934/cpaa.2013.12.99

[14]

Fouad Hadj Selem, Hiroaki Kikuchi, Juncheng Wei. Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4613-4626. doi: 10.3934/dcds.2013.33.4613

[15]

Xing-Bin Pan. An eigenvalue variation problem of magnetic Schrödinger operator in three dimensions. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 933-978. doi: 10.3934/dcds.2009.24.933

[16]

Ihyeok Seo. Carleman estimates for the Schrödinger operator and applications to unique continuation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1013-1036. doi: 10.3934/cpaa.2012.11.1013

[17]

Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems & Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169

[18]

Joel Andersson, Leo Tzou. Stability for a magnetic Schrödinger operator on a Riemann surface with boundary. Inverse Problems & Imaging, 2018, 12 (1) : 1-28. doi: 10.3934/ipi.2018001

[19]

Ru-Yu Lai. Global uniqueness for an inverse problem for the magnetic Schrödinger operator. Inverse Problems & Imaging, 2011, 5 (1) : 59-73. doi: 10.3934/ipi.2011.5.59

[20]

Leyter Potenciano-Machado, Alberto Ruiz. Stability estimates for a magnetic Schrödinger operator with partial data. Inverse Problems & Imaging, 2018, 12 (6) : 1309-1342. doi: 10.3934/ipi.2018055

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (9)
  • HTML views (1)
  • Cited by (0)

Other articles
by authors

[Back to Top]