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A Hopf lemma and regularity for fractional \(p \)-Laplacians. (English) Zbl 1439.35523

Summary: In this paper, we study qualitative properties of the fractional \(p\)-Laplacian. Specifically, we establish a Hopf type lemma for positive weak super-solutions of the fractional \(p-\) Laplacian equation with Dirichlet condition. Moreover, an optimal condition is obtained to ensure \((-\triangle)_p^s u\in C^1(\mathbb{R}^n)\) for smooth functions \(u\).

MSC:

35R11 Fractional partial differential equations
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35B65 Smoothness and regularity of solutions to PDEs
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[1] G. Alberti; G. Bellettini, A nonlocal anisotropicmodel for phase transitions Ⅰ: The optimal profile problem, Math. Ann., 310, 527-560 (1998) · Zbl 0891.49021
[2] C. Bjorland; L. Caffarelli; A. Figalli, Non-local gradient dependent operators, Adv. Math., 230, 1859-1894 (2012) · Zbl 1252.35099
[3] C. Bjorland; L. Caffarelli; A. Figalli, Nonlocal tug-of-war and the infinity fractional Laplacian, Comm. Pure Appl. Math., 65, 337-380 (2012) · Zbl 1235.35278
[4] C. Brandle; E. Colorado; A. de Pablo; U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143, 39-71 (2013) · Zbl 1290.35304
[5] L. Brasco; E. Lindgren; A. Schikorra, Higher \(H \ddot{o}\) der regularity for the fractional p-Laplacian in the superquadratic case, Adv. Math., 338, 782-846 (2018) · Zbl 1400.35049
[6] K. Bogdan; T. Grzywny; M. Ryznar, Heat kernel estimates for the fractional Laplacian with Dirichlet conditions, Ann. Probab., 38, 1901-1923 (2010) · Zbl 1204.60074
[7] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20, Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016.
[8] L. Caffarelli; L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32, 1245-1260 (2007) · Zbl 1143.26002
[9] W. Chen; Y. Fang; R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274, 167-198 (2015) · Zbl 1372.35332
[10] W. Chen; C. Li, Maximum principles for the fractional \(p\)-Laplacian and symmetry of solutions, Adv. Math., 335, 735-758 (2018) · Zbl 1395.35055
[11] W. Chen; Y. Li; R. Zhang, A direct method of moving spheres on fractional order equations, J. Funct. Anal., 272, 4131-4157 (2017) · Zbl 1431.35225
[12] W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differential Equations, 56 (2017), Art. 29, 18 pp. · Zbl 1368.35110
[13] W. Chen; C. Li; Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math., 308, 404-437 (2017) · Zbl 1362.35320
[14] W. Chen, C. Li and Y. Li, A direct blowing-up and rescaling argument on nonlocal elliptic equations, Internat. J. Math.,27 (2016), 1650064, 20 pp. · Zbl 1348.35080
[15] W. Chen, Y. Li and P. Ma, The Fractional Laplacian, A book to be published by World Scientific Publishing C, 2019.
[16] W. Chen; S. Qi, Direct methods on fractional equations, Discrete Contin. Dyn. Syst., 39, 1269-1310 (2019) · Zbl 1408.35038
[17] W. Chen; J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations, 260, 4758-4785 (2016) · Zbl 1336.35089
[18] A. Di Castro; T. Kuusi; G. Palatucci, Local behavior of fractional p-minimizers, Ann. Inst. H. Poincare Anal. Non Lineaire, 33, 1279-1299 (2016) · Zbl 1355.35192
[19] E. Di Nezza; G. Palatucci; E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136, 521-573 (2012) · Zbl 1252.46023
[20] M. M. Fall; S. Jarohs, Overdetermined problems with fractional Laplacian, ESAIM Control Optim. Calc. Var., 21, 924-938 (2015) · Zbl 1329.35223
[21] A. Greco; R. Servadei, Hopf’s lemma and constrained radial symmetry for the fractional Laplacian, Math. Res. Lett., 23, 863-885 (2016) · Zbl 1361.35195
[22] A. Iannizzotto; S. Mosconi; M. Squassina, Global \(H \ddot{o}\) der regularity for the fractional \(p\)-Laplacian, Rev. Mat. Iberoam., 32, 1353-1392 (2016) · Zbl 1433.35447
[23] H. Ishii; G. Nakamura, A class of integral equations and approximation of \(p\)-Laplace equations, Calc. Var. Partial Differential Equations, 37, 485-522 (2010) · Zbl 1198.45005
[24] L. Jin; Y. Li, A Hopf’s Lemma and the Boundary Regularity for the Fractional \(p\)-Laplacian, Discrete Contin. Dyn. Syst., 39, 1477-1495 (2019) · Zbl 1439.35598
[25] C. Li; W. Chen, A Hopf type lemma for fractional equations, Proc. Amer. Math. Soc., 147, 1565-1575 (2019) · Zbl 1417.35034
[26] S. Mosconi; M. Squassina, Recent progresses in the theory of nonlinear nonlocal problems, Bruno Pini Math. Analysis Sem., 7, 147-164 (2016) · Zbl 1371.35329
[27] X. Ros-Oton; J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101, 275-302 (2014) · Zbl 1285.35020
[28] X. Ros-Oton; E. Valdinoci, The Dirichlet problem for nonlocal operators with singular kernels: convex and nonconvex domains, Adv. Math., 288, 732-790 (2016) · Zbl 1334.35397
[29] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60, 67-112 (2007) · Zbl 1141.49035
[30] Y. Sire; E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal., 256, 1842-1864 (2009) · Zbl 1163.35019
[31] Y. Sire; E. Valdinoci, Rigidity results for some boundary quasilinear phase transitions, Comm. Partial Differential Equations, 34, 765-784 (2009) · Zbl 1188.35091
[32] E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl., 49, 33-44 (2009) · Zbl 1242.60047
[33] R. Zhuo; W. Chen; X. Cui; Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36, 1125-1141 (2016) · Zbl 1322.31007
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