A Harnack inequality for fractional Laplace equations with lower order terms

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September 2011, 31(3): 975-983. doi: 10.3934/dcds.2011.31.975

A Harnack inequality for fractional Laplace equations with lower order terms

Jinggang Tan ^1, and Jingang Xiong ^2,

1.	Departamento de Matemática, Universidad Técnica Federico Santa María, Avda. España 1680, Valparaíso, Chile
2.	School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

Received February 2010 Revised May 2011 Published August 2011

Abstract
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We establish a Harnack inequality of fractional Laplace equations without imposing sign condition on the coefficient of zero order term via the Moser's iteration and John-Nirenberg inequality.

Keywords: Harnack inequality, Moser's iteration., Fractional Laplace equations.

Mathematics Subject Classification: Primary: 35J70, 35B45, 35S05, 35J2.

Citation: Jinggang Tan, Jingang Xiong. A Harnack inequality for fractional Laplace equations with lower order terms. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 975-983. doi: 10.3934/dcds.2011.31.975

References:

[1]	R. F. Bass and D. A. Levin, Harnack inequalities for jump processes, Potential Anal., 17 (2002), 375-388 doi: 10.1023/A:1016378210944.
[2]	X. Cabre and Y. Sire, Nonlinear equations for fractional laplacians I: Regularity, maximum principles, and hamiltonian estimates, preprint, arXiv:1012.0867v1.
[3]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
[4]	Z.-Q. Chen and R. Song, Estimates on Green functions and Poisson kernels for symmetric stable processes, Math. Ann., 312 (1998), 465-501. doi: 10.1007/s002080050232.
[5]	E. Fabes, C. Kenig and R. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations, 7 (1982), 77-116.
[6]	Q. Han and F.-H. Lin, "Elliptic Partial Differential Equations," Courant Lecture Notes in Mathematics, 1, New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 1997.
[7]	Z.-C. Han and Y. Y. Li, The Yamabe problem on manifolds with boundary: Existence and compactness results, Duke Math. J., 99 (1999), 489-542. doi: 10.1215/S0012-7094-99-09916-7.
[8]	F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math., 14 (1961), 415-426. doi: 10.1002/cpa.3160140317.
[9]	M. Kassmann, The classical Harnack inequality fails for non-local operators, preprint.
[10]	B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., 192 (1974), 261-274. doi: 10.1090/S0002-9947-1974-0340523-6.
[11]	E. Stein, "Singular Integrals and Differentiability Properties of Function," Princeton Mathematical Series, 30, Princeton University Press, Princeton, NJ, 1970.

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References:

[1]	R. F. Bass and D. A. Levin, Harnack inequalities for jump processes, Potential Anal., 17 (2002), 375-388 doi: 10.1023/A:1016378210944.
[2]	X. Cabre and Y. Sire, Nonlinear equations for fractional laplacians I: Regularity, maximum principles, and hamiltonian estimates, preprint, arXiv:1012.0867v1.
[3]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
[4]	Z.-Q. Chen and R. Song, Estimates on Green functions and Poisson kernels for symmetric stable processes, Math. Ann., 312 (1998), 465-501. doi: 10.1007/s002080050232.
[5]	E. Fabes, C. Kenig and R. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations, 7 (1982), 77-116.
[6]	Q. Han and F.-H. Lin, "Elliptic Partial Differential Equations," Courant Lecture Notes in Mathematics, 1, New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 1997.
[7]	Z.-C. Han and Y. Y. Li, The Yamabe problem on manifolds with boundary: Existence and compactness results, Duke Math. J., 99 (1999), 489-542. doi: 10.1215/S0012-7094-99-09916-7.
[8]	F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math., 14 (1961), 415-426. doi: 10.1002/cpa.3160140317.
[9]	M. Kassmann, The classical Harnack inequality fails for non-local operators, preprint.
[10]	B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., 192 (1974), 261-274. doi: 10.1090/S0002-9947-1974-0340523-6.
[11]	E. Stein, "Singular Integrals and Differentiability Properties of Function," Princeton Mathematical Series, 30, Princeton University Press, Princeton, NJ, 1970.

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