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

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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

Abstract
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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 and 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, Inc., New York; National Bureau of Standards, Washington, DC, 1984.
[2]	R. M. Blumenthal, R. K. Getoor and D. B. Ray, On the distribution of first hits for the symmetric stable processes, Trans. Amer. Math. Soc., 99 (1961), 540-554.
[3]	Claudia Bucur and Enrico Valdinoci, Nonlocal diffusion and applications, accepted for Publication for the Springer Series Lecture Notes of the Unione Matematica Italiana, preprint arXiv:1504.08292, 2015.
[4]	Eleonora Di Nezza, Giampiero Palatucci and Enrico Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[5]	Bartłomiej Dyda, Fractional Hardy inequality with a remainder term, Colloq. Math., 122 (2011), 59-67. doi: 10.4064/cm122-1-6.
[6]	Bartłomiej Dyda, Fractional calculus for power functions and eigenvalues of the fractional Laplacian, Fract. Calc. Appl. Anal., 15 (2012), 536-555. doi: 10.2478/s13540-012-0038-8.
[7]	Lawrence C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, second edition, 2010. doi: 10.1090/gsm/019.
[8]	Yitzhak Katznelson, An Introduction to Harmonic Analysis, Cambridge Mathematical Library. Cambridge University Press, Cambridge, third edition, 2004. doi: 10.1017/CBO9781139165372.
[9]	Tadeusz Kulczycki, Properties of Green function of symmetric stable processes, Probab. Math. Statist., 17 (1997), 339-364.
[10]	N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180.
[11]	Michael Reed and Barry Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, New York-London, 1972.
[12]	Luis Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153.
[13]	Richard L. Wheeden and Antoni Zygmund, Measure and Integral, An introduction to real analysis, Pure and Applied Mathematics, Vol. 43, Marcel Dekker, Inc., New York-Basel, 1977.

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, Inc., New York; National Bureau of Standards, Washington, DC, 1984.
[2]	R. M. Blumenthal, R. K. Getoor and D. B. Ray, On the distribution of first hits for the symmetric stable processes, Trans. Amer. Math. Soc., 99 (1961), 540-554.
[3]	Claudia Bucur and Enrico Valdinoci, Nonlocal diffusion and applications, accepted for Publication for the Springer Series Lecture Notes of the Unione Matematica Italiana, preprint arXiv:1504.08292, 2015.
[4]	Eleonora Di Nezza, Giampiero Palatucci and Enrico Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[5]	Bartłomiej Dyda, Fractional Hardy inequality with a remainder term, Colloq. Math., 122 (2011), 59-67. doi: 10.4064/cm122-1-6.
[6]	Bartłomiej Dyda, Fractional calculus for power functions and eigenvalues of the fractional Laplacian, Fract. Calc. Appl. Anal., 15 (2012), 536-555. doi: 10.2478/s13540-012-0038-8.
[7]	Lawrence C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, second edition, 2010. doi: 10.1090/gsm/019.
[8]	Yitzhak Katznelson, An Introduction to Harmonic Analysis, Cambridge Mathematical Library. Cambridge University Press, Cambridge, third edition, 2004. doi: 10.1017/CBO9781139165372.
[9]	Tadeusz Kulczycki, Properties of Green function of symmetric stable processes, Probab. Math. Statist., 17 (1997), 339-364.
[10]	N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180.
[11]	Michael Reed and Barry Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, New York-London, 1972.
[12]	Luis Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153.
[13]	Richard L. Wheeden and Antoni Zygmund, Measure and Integral, An introduction to real analysis, Pure and Applied Mathematics, Vol. 43, Marcel Dekker, Inc., New York-Basel, 1977.

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