A Harnack inequality for fractional Laplace equations with lower order terms

Jinggang Tan; Jingang Xiong

doi:10.3934/dcds.2011.31.975

Article Contents

2011, Volume 31, Issue 3: 975-983. Doi: 10.3934/dcds.2011.31.975

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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
2.
School of Mathematical Sciences, Beijing Normal University, Beijing 100875

Received: February 2010

Revised: May 2011

Published: September 2011

Abstract / Introduction Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 35J70, 35B45, 35S05, 35J25.

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References

[1]	R. F. Bass and D. A. Levin, Harnack inequalities for jump processes, Potential Anal., 17 (2002), 375-388doi: 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.