Some monotonicity results for the fractional Laplacian in unbounded domain

References

• Bouchard JP, Georges A. Anomalous diffusion in disordered media, statistical mechanics, models and physical applications. Phys Rep. 1990;195:127–293. doi: 10.1016/0370-1573(90)90099-N
   Web of Science ®Google Scholar
• Constantin P. Euler equations, Navier–Stokes equations and turbulence. Mathematical Foundation of Turbulent Viscous Flows, Vol. 1871 of Lecture Notes in Math. 1C43, Berlin: Springer; 2006.
   Google Scholar
• Caffarelli L, Vasseur L. Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann Math. 2010;171:1903–1930. doi: 10.4007/annals.2010.171.1903
   Web of Science ®Google Scholar
• Tarasov V, Zaslasvky G. Fractional dynamics of systems with long-range interaction. Commun Nonlinear Sci Numer Simul. 2006;11:885–889. doi: 10.1016/j.cnsns.2006.03.005
   Google Scholar
• Applebaum D. Lévy processes and stochastic calculus. 2nd ed. Cambridge: Cambridge University Press; 2009. (Cambridge Studies in Advance Mathematics. 116).
   Google Scholar
• Bertoin J. Lévy processes. Cambridge: Cambridge University Press; 1996. (Cambridge Tracts in Mathematics, 121).
   Google Scholar
• Bucur C, Valdinoci E. Nonlocal diffusion and applications. Bologna: Springer, Cham; Unione Matematica Italiana, 2016. (Lecture Notes of the Unione Matematica Italiana, 20).
   Google Scholar
• Chang S, González M. Fractional Laplacian in conformal geometry. Adv Math. 2011;226:1410–1432. doi: 10.1016/j.aim.2010.07.016
   Web of Science ®Google Scholar
• Caffarelli L, Silvestre L. An extension problem related to the fractional Laplacian. Comm PDE. 2007;32:1245–1260. doi: 10.1080/03605300600987306
   Web of Science ®Google Scholar
• Chen W, Fang Y, Yang R. Liouville theorems involving the fractional Laplacian on a half space. Adv Math. 2015;274:167–198. doi: 10.1016/j.aim.2014.12.013
   Web of Science ®Google Scholar
• Chen W, Li C, Ou B. Classification of solutions for an integral equation. Comm Pure Appl Math. 2006;59:330–343. doi: 10.1002/cpa.20116
   Web of Science ®Google Scholar
• Chen W, Li C, Ou B. Qualitative properties of solutions for an integral equation. Disc Cont Dyn Sys. 2005;12:347–354. doi: 10.3934/dcds.2005.12.347
   Web of Science ®Google Scholar
• Fall M, Jarohs S. Overdetermined problems with fractional Laplacian. ESAIM Control Optim Calc Var. 2015;21:924–938. doi: 10.1051/cocv/2014048
   Web of Science ®Google Scholar
• Barrios B, Montoro L, Sciunzi B. On the moving plane method for nonlocal problems in bounded domains. J Anal Math. 2018;1:37–57. doi: 10.1007/s11854-018-0031-1
   Web of Science ®Google Scholar
• Chen W, Li C, Li Y. A direct method of moving planes for the fractional Laplacian. Adv Math. 2017;308:404–437. doi: 10.1016/j.aim.2016.11.038
   Web of Science ®Google Scholar
• Chen W, Li Y, Ma P. The fractional Laplacian. World Scientific Publishing Co.; June 2019.
   Google Scholar
• Dipierro S, Soave N, Valdinoci E. On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results. Math Ann. 2017;369:1283–1326. doi: 10.1007/s00208-016-1487-x
   Web of Science ®Google Scholar
• Wu L, Yu M. Some monotonicity results for the fractional p-Laplacian in unbounded domains. preprint, 2019.
   Google Scholar
• Ghoussoub N, Gui C. On a conjecture of De Giorgi and some related problems. Math Ann. 1998;311:481–491. doi: 10.1007/s002080050196
   Web of Science ®Google Scholar
• Cabré X, Sire Y. Nonlinear equations for fractional Laplacians, I: regularity, maximum principles, and Hamiltonian estimates. Ann Inst H Poincaré Anal Non Linéaire. 2014;31:23–53. doi: 10.1016/j.anihpc.2013.02.001
   Web of Science ®Google Scholar
• Sire Y, Valdinoci E. Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result. J Funct Anal. 2009;256:1842–1864. doi: 10.1016/j.jfa.2009.01.020
   Web of Science ®Google Scholar
• Cabré X, Solà-Morales J. Layer solutions in a half-space for boundary reactions. Comm Pure Appl Math. 2005;58:1678–1732. doi: 10.1002/cpa.20093
   Web of Science ®Google Scholar
• Cabré X, Cinti E. Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian. Discrete Contin Dyn Syst. 2010;28:1179–1206. doi: 10.3934/dcds.2010.28.1179
   Web of Science ®Google Scholar
• Cabré X, Cinti E. Sharp energy estimates for nonlinear fractional diffusion equations. Calc Var Partial Differ Equ. 2014;49:233–269. doi: 10.1007/s00526-012-0580-6
   Web of Science ®Google Scholar
• Savin O, Valdinoci E. Some monotonicity results for minimizers in the calculus of variations. J Funct Anal. 2013;264:2469–2496. doi: 10.1016/j.jfa.2013.02.005
   Web of Science ®Google Scholar
• Dipierro S, Miraglio P, Valdinoci E. Symmetry results for the solutions of a partial differential equation arising in water waves, 2019 MATRIX annals, 153–166, MATRIX Book Ser., Springer, Cham, 2019.
   Google Scholar
• Dipierro S, Farina A, Valdinoci E. A three-dimensional symmetry result for a phase transition equation in the genuinely nonlocal regime. Calc Var Partial Differ Equ. 2018;57:15. doi: 10.1007/s00526-017-1295-5
   Web of Science ®Google Scholar
• Dipierro S, Serra J, Valdinoci E. Improvement of flatness for nonlocal phase transitions. Amer J Math. in print.
   Google Scholar
• Dipierro S, Serra J, Valdinoci E. Nonlocal phase transitions: rigidity results and anisotropic geometry. Rend Semin Mat Univ Politec Torino. 2016;74:135–149.
   Google Scholar
• Savin O. Rigidity of minimizers in nonlocal phase transitions. Anal PDE. 2018;11:1881–1900. doi: 10.2140/apde.2018.11.1881
   Web of Science ®Google Scholar
• Savin O. Rigidity of minimizers in nonlocal phase transitions II. Anal Theory Appl. 2019;35:1–27. doi: 10.4208/ata.OA-0008
   Google Scholar
• Farina A, Valdinoci E. Rigidity results for elliptic PDEs with uniform limits: an abstract framework with applications. Indiana Univ Math J. 2011;60:121–141. doi: 10.1512/iumj.2011.60.4433
   Web of Science ®Google Scholar
• Chen W, Hu Y. Monotonicity of positive solutions for nonlocal problems in unbounded domains. preprint, 2019.
   Google Scholar
• Li Z, Zhang Q. Subsolutions and Hopf's theorem of fractional p-Laplacian, arXiv: 1905.00127, 2019.
   Google Scholar
• Wu L, Chen W. Monotonicity of solutions for fractional equations with De Giorgi type nonlinearities, arXiv: 1905.09999, 2019.
   Google Scholar