American Institute of Mathematical Sciences

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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References:
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