July  2013, 33(7): 3153-3170. doi: 10.3934/dcds.2013.33.3153

Harnack's inequality for fractional nonlocal equations

1. 

Department of Mathematics, The University of Texas at Austin, 1 University Station, C1200, Austin, TX 78712-1202, United States

2. 

Department of Mathematics, Sun Yat-sen (Zhongshan) University, 510275 Guangzhou, China

Received  March 2012 Revised  June 2012 Published  January 2013

We prove interior Harnack's inequalities for solutions of fractional nonlocal equations. Our examples include fractional powers of divergence form elliptic operators with potentials, operators arising in classical orthogonal expansions and the radial Laplacian. To get the results we use an analytic method based on a generalization of the Caffarelli--Silvestre extension problem, the Harnack's inequality for degenerate Schrödinger operators proved by C. E. Gutiérrez, and a transference method. In this manner we apply local PDE techniques to nonlocal operators. On the way a maximum principle and a Liouville theorem for some fractional nonlocal equations are obtained.
Citation: Pablo Raúl Stinga, Chao Zhang. Harnack's inequality for fractional nonlocal equations. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 3153-3170. doi: 10.3934/dcds.2013.33.3153
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show all references

References:
[1]

Proc. Indian Acad. Sci. Math. Sci., 119 (2009), 203-220. doi: 10.1007/s12044-009-0021-4.  Google Scholar

[2]

Glasgow Math. J., 48 (2006), 203-215. doi: 10.1017/S0017089506003004.  Google Scholar

[3]

J. Anal. Math., 107 (2009), 195-219. doi: 10.1007/s11854-009-0008-1.  Google Scholar

[4]

in "Proceedings of the Conference on Differential Equations (dedicated to A. Weinstein)", University of Maryland Book Store, College Park, Md., (1956), 23-48.  Google Scholar

[5]

in "ICIAM 07-6th International Congress on Industrial and Applied Mathematics," Eur. Math. Soc., Zürich, (2009), 43-56. doi: 10.4171/056-1/3.  Google Scholar

[6]

Invent. Math., 171 (2008), 425-461. doi: 10.1007/s00222-007-0086-6.  Google Scholar

[7]

Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.  Google Scholar

[8]

Ann. of Math. (2), 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.  Google Scholar

[9]

Cambridge Tracts in Mathematics 92, Cambridge Univ. Press, Cambridge, 1989. doi: 10.1017/CBO9780511566158.  Google Scholar

[10]

Comm. Partial Differential Equations, 7 (1982), 77-116. doi: 10.1080/03605308208820218.  Google Scholar

[11]

Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[12]

Trans. Amer. Math. Soc., 312 (1989), 403-419. doi: 10.2307/2001222.  Google Scholar

[13]

J. Funct. Anal., 120 (1994), 107-134. doi: 10.1006/jfan.1994.1026.  Google Scholar

[14]

Houston J. Math., 27 (2001), 579-592.  Google Scholar

[15]

preprint (2011), 59 pp, Google Scholar

[16]

Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[17]

Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965.  Google Scholar

[18]

Trans. Amer. Math. Soc., 118 (1965), 17-92.  Google Scholar

[19]

preprint (2012), 16 pp, Google Scholar

[20]

McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973.  Google Scholar

[21]

Nature, 363 (1993), 31-37. Google Scholar

[22]

Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153.  Google Scholar

[23]

Comm. Partial Differential Equations, 35 (2010), 2092-2122. doi: 10.1080/03605301003735680.  Google Scholar

[24]

Fourth edition, American Mathematical Society Colloquium Publications XXIII, American Mathematical Society, Providence, R.I., 1975.  Google Scholar

[25]

Discrete Contin. Dyn. Syst., 31 (2011), 975-983. doi: 10.3934/dcds.2011.31.975.  Google Scholar

[26]

Mathematical Notes 42, Princeton University Press, Princeton, NJ, 1993.  Google Scholar

[27]

Third edition, Chelsea Publishing Co., New York, 1986.  Google Scholar

[28]

Ann. Scuola Norm. Sup. Pisa (3), 27 (1973), 265-308.  Google Scholar

[29]

Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar

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