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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

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 & 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.  Google Scholar

[2]

X. Cabre and Y. Sire, Nonlinear equations for fractional laplacians I: Regularity, maximum principles, and hamiltonian estimates,, preprint, ().   Google Scholar

[3]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

M. Kassmann, The classical Harnack inequality fails for non-local operators,, preprint., ().   Google Scholar

[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.  Google Scholar

[11]

E. Stein, "Singular Integrals and Differentiability Properties of Function," Princeton Mathematical Series, 30, Princeton University Press, Princeton, NJ, 1970.  Google Scholar

show all references

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.  Google Scholar

[2]

X. Cabre and Y. Sire, Nonlinear equations for fractional laplacians I: Regularity, maximum principles, and hamiltonian estimates,, preprint, ().   Google Scholar

[3]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

M. Kassmann, The classical Harnack inequality fails for non-local operators,, preprint., ().   Google Scholar

[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.  Google Scholar

[11]

E. Stein, "Singular Integrals and Differentiability Properties of Function," Princeton Mathematical Series, 30, Princeton University Press, Princeton, NJ, 1970.  Google Scholar

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