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]

Fausto Ferrari. Mean value properties of fractional second order operators. Communications on Pure & Applied Analysis, 2015, 14 (1) : 83-106. doi: 10.3934/cpaa.2015.14.83
[2]

Jeremiah Birrell. A posteriori error bounds for two point boundary value problems: A green's function approach. Journal of Computational Dynamics, 2015, 2 (2) : 143-164. doi: 10.3934/jcd.2015001
[3]

Xi Wang, Zuhan Liu, Ling Zhou. Asymptotic decay for the classical solution of the chemotaxis system with fractional Laplacian in high dimensions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 4003-4020. doi: 10.3934/dcdsb.2018121
[4]

Juan Li, Wenqiang Li. Controlled reflected mean-field backward stochastic differential equations coupled with value function and related PDEs. Mathematical Control & Related Fields, 2015, 5 (3) : 501-516. doi: 10.3934/mcrf.2015.5.501
[5]

Kyoungsun Kim, Gen Nakamura, Mourad Sini. The Green function of the interior transmission problem and its applications. Inverse Problems & Imaging, 2012, 6 (3) : 487-521. doi: 10.3934/ipi.2012.6.487
[6]

Jongkeun Choi, Ki-Ahm Lee. The Green function for the Stokes system with measurable coefficients. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1989-2022. doi: 10.3934/cpaa.2017098
[7]

Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks & Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007
[8]

Virginia Agostiniani, Rolando Magnanini. Symmetries in an overdetermined problem for the Green's function. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 791-800. doi: 10.3934/dcdss.2011.4.791
[9]

Sungwon Cho. Alternative proof for the existence of Green's function. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1307-1314. doi: 10.3934/cpaa.2011.10.1307
[10]

Zhi-Min Chen. Straightforward approximation of the translating and pulsating free surface Green function. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2767-2783. doi: 10.3934/dcdsb.2014.19.2767
[11]

Manh Hong Duong, Hoang Minh Tran. On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3407-3438. doi: 10.3934/dcds.2018146
[12]

Yutong Chen, Jiabao Su. Resonant problems for fractional Laplacian. Communications on Pure & Applied Analysis, 2017, 16 (1) : 163-188. doi: 10.3934/cpaa.2017008
[13]

José G. Llorente. Mean value properties and unique continuation. Communications on Pure & Applied Analysis, 2015, 14 (1) : 185-199. doi: 10.3934/cpaa.2015.14.185
[14]

Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285
[15]

M. P. de Oliveira. On 3-graded Lie algebras, Jordan pairs and the canonical kernel function. Electronic Research Announcements, 2003, 9: 142-151.

[16]

Mourad Choulli. Local boundedness property for parabolic BVP's and the Gaussian upper bound for their Green functions. Evolution Equations & Control Theory, 2015, 4 (1) : 61-67. doi: 10.3934/eect.2015.4.61
[17]

Maoding Zhen, Jinchun He, Haoyun Xu. Critical system involving fractional Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (1) : 237-253. doi: 10.3934/cpaa.2019013
[18]

Mikko Kemppainen, Peter Sjögren, José Luis Torrea. Wave extension problem for the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4905-4929. doi: 10.3934/dcds.2015.35.4905
[19]

Vladimir E. Fedorov, Natalia D. Ivanova. Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 687-696. doi: 10.3934/dcdss.2016022
[20]

Ugo Bessi. The stochastic value function in metric measure spaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1819-1839. doi: 10.3934/dcds.2017076

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences