American Institute of Mathematical Sciences

Advanced
March  2016, 15(2): 657-699. doi: 10.3934/cpaa.2016.15.657

Some observations on the Green function for the ball in the fractional Laplace framework

Claudia Bucur 1,

1. 

Dipartimento di Matematica "Federigo Enriques", Università degli Studi di Milano, Via Cesare Saldini, 50, I-20133, Milano, Italy

Received  March 2015 Revised  November 2015 Published  January 2016

We consider a fractional Laplace equation and we give a self-contained elementary exposition of the representation formula for the Green function on the ball. In this exposition, only elementary calculus techniques will be used, in particular, no probabilistic methods or computer assisted algebraic manipulations are needed. The main result in itself is not new (see for instance [2, 9]), however we believe that the exposition is original and easy to follow, hence we hope that this paper will be accessible to a wide audience of young researchers and graduate students that want to approach the subject, and even to professors that would like to present a complete proof in a PhD or Master Degree course.
Keywords: Poisson kernel, fundamental solution, Fractional Laplacian, Green function, mean value property..
Mathematics Subject Classification: Primary: 35C15, 35S05; Secondary: 35S30, 31B1.
Citation: Claudia Bucur. Some observations on the Green function for the ball in the fractional Laplace framework. Communications on Pure & Applied Analysis, 2016, 15 (2) : 657-699. doi: 10.3934/cpaa.2016.15.657
References:
[1]

Milton Abramowitz and Irene Anne Stegun eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,, A Wiley-Interscience Publication. John Wiley & Sons, (1984).   Google Scholar

[2]

R. M. Blumenthal, R. K. Getoor and D. B. Ray, On the distribution of first hits for the symmetric stable processes,, \emph{Trans. Amer. Math. Soc.}, 99 (1961), 540.   Google Scholar

[3]

Claudia Bucur and Enrico Valdinoci, Nonlocal diffusion and applications,, accepted for Publication for the Springer Series \emph{Lecture Notes of the Unione Matematica Italiana}, (2015).   Google Scholar

[4]

Eleonora Di Nezza, Giampiero Palatucci and Enrico Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, \emph{Bull. Sci. Math.}, 136 (2012), 521.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[5]

Bartłomiej Dyda, Fractional Hardy inequality with a remainder term,, \emph{Colloq. Math.}, 122 (2011), 59.  doi: 10.4064/cm122-1-6.  Google Scholar

[6]

Bartłomiej Dyda, Fractional calculus for power functions and eigenvalues of the fractional Laplacian,, \emph{Fract. Calc. Appl. Anal.}, 15 (2012), 536.  doi: 10.2478/s13540-012-0038-8.  Google Scholar

[7]

Lawrence C. Evans, Partial Differential Equations,, volume 19 of Graduate Studies in Mathematics, (2010).  doi: 10.1090/gsm/019.  Google Scholar

[8]

Yitzhak Katznelson, An Introduction to Harmonic Analysis,, Cambridge Mathematical Library. Cambridge University Press, (2004).  doi: 10.1017/CBO9781139165372.  Google Scholar

[9]

Tadeusz Kulczycki, Properties of Green function of symmetric stable processes,, \emph{Probab. Math. Statist.}, 17 (1997), 339.   Google Scholar

[10]

N. S. Landkof, Foundations of Modern Potential Theory,, Springer-Verlag, (1972).   Google Scholar

[11]

Michael Reed and Barry Simon, Methods of Modern Mathematical Physics. I. Functional Analysis,, Academic Press, (1972).   Google Scholar

[12]

Luis Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator,, \emph{Comm. Pure Appl. Math.}, 60 (2007), 67.  doi: 10.1002/cpa.20153.  Google Scholar

[13]

Richard L. Wheeden and Antoni Zygmund, Measure and Integral,, An introduction to real analysis, (1977).   Google Scholar

show all references

References:
[1]

Milton Abramowitz and Irene Anne Stegun eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,, A Wiley-Interscience Publication. John Wiley & Sons, (1984).   Google Scholar

[2]

R. M. Blumenthal, R. K. Getoor and D. B. Ray, On the distribution of first hits for the symmetric stable processes,, \emph{Trans. Amer. Math. Soc.}, 99 (1961), 540.   Google Scholar

[3]

Claudia Bucur and Enrico Valdinoci, Nonlocal diffusion and applications,, accepted for Publication for the Springer Series \emph{Lecture Notes of the Unione Matematica Italiana}, (2015).   Google Scholar

[4]

Eleonora Di Nezza, Giampiero Palatucci and Enrico Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, \emph{Bull. Sci. Math.}, 136 (2012), 521.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[5]

Bartłomiej Dyda, Fractional Hardy inequality with a remainder term,, \emph{Colloq. Math.}, 122 (2011), 59.  doi: 10.4064/cm122-1-6.  Google Scholar

[6]

Bartłomiej Dyda, Fractional calculus for power functions and eigenvalues of the fractional Laplacian,, \emph{Fract. Calc. Appl. Anal.}, 15 (2012), 536.  doi: 10.2478/s13540-012-0038-8.  Google Scholar

[7]

Lawrence C. Evans, Partial Differential Equations,, volume 19 of Graduate Studies in Mathematics, (2010).  doi: 10.1090/gsm/019.  Google Scholar

[8]

Yitzhak Katznelson, An Introduction to Harmonic Analysis,, Cambridge Mathematical Library. Cambridge University Press, (2004).  doi: 10.1017/CBO9781139165372.  Google Scholar

[9]

Tadeusz Kulczycki, Properties of Green function of symmetric stable processes,, \emph{Probab. Math. Statist.}, 17 (1997), 339.   Google Scholar

[10]

N. S. Landkof, Foundations of Modern Potential Theory,, Springer-Verlag, (1972).   Google Scholar

[11]

Michael Reed and Barry Simon, Methods of Modern Mathematical Physics. I. Functional Analysis,, Academic Press, (1972).   Google Scholar

[12]

Luis Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator,, \emph{Comm. Pure Appl. Math.}, 60 (2007), 67.  doi: 10.1002/cpa.20153.  Google Scholar

[13]

Richard L. Wheeden and Antoni Zygmund, Measure and Integral,, An introduction to real analysis, (1977).   Google Scholar

[1]

Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443
[2]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137
[3]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052
[4]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442
[5]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445
[6]

Jie Li, Xiangdong Ye, Tao Yu. Mean equicontinuity, complexity and applications. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 359-393. doi: 10.3934/dcds.2020167
[7]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469
[8]

Xin Guo, Lexin Li, Qiang Wu. Modeling interactive components by coordinate kernel polynomial models. Mathematical Foundations of Computing, 2020, 3 (4) : 263-277. doi: 10.3934/mfc.2020010
[9]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296
[10]

Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020071
[11]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248
[12]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447
[13]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075
[14]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348
[15]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440
[16]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118
[17]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016
[18]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070
[19]

Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017
[20]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (148)
  • HTML views (0)
  • Cited by (27)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences