Global Hölder regularity for the fractional \(p\)-Laplacian. (English) Zbl 1433.35447

Summary: By virtue of barrier arguments we prove \(C^\alpha\)-regularity up to the boundary for the weak solutions of a non-local, non-linear problem driven by the fractional \(p\)-Laplacian operator. The equation is boundedly inhomogeneous and the boundary conditions are of Dirichlet type. We employ different methods according to the singular \((p<2)\) of degenerate \((p>2)\) case.


35R11 Fractional partial differential equations
35B65 Smoothness and regularity of solutions to PDEs
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
47G20 Integro-differential operators
Full Text: DOI arXiv


[1] Aikawa, H., Kilpel”ainen, T., Shanmugalingam, N. and Zhong, X.:Boundary Harnack principle forp-harmonic functions in smooth Euclidean domains. Potential Anal.26 (2007), no. 3, 281–301. · Zbl 1121.35060
[2] Baernstein, A. II: A unified approach to symmetrization. In Partial differential equations of elliptic type (Cortona, 1992), 47–91. Symposia Mathematica 35, Cambridge University Press, Cambridge, 1994.
[3] Bjorland, C., Caffarelli, L. and Figalli, A.: Non-local gradient dependent operators. Adv. Math.230 (2012), no. 4-6, 1859–1894. · Zbl 1252.35099
[4] Cabr’e, X. and Sire, Y.:Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates. Ann. Inst. H. Poincar’e Anal. Non Lin’eaire31 (2014), no. 1, 23–53. · Zbl 1286.35248
[5] Caffarelli, L. and Silvestre, L.: Regularity theory for fully nonlinear integrodifferential equations. Comm. Pure Appl. Math.62 (2009), no. 5, 597–638. · Zbl 1170.45006
[6] Caffarelli, L. and Silvestre, L.: Regularity results for nonlocal equations by approximation. Arch. Ration. Mech. Anal.200 (2011), no. 1, 59–88. · Zbl 1231.35284
[7] Di Castro, A., Kuusi, T. and Palatucci, G.: Local behavior of fractionalpminimizers. Ann. Inst. H. Poincar’e Anal. Non Lin’eaire33 (2016), no. 5, 1279–1299. · Zbl 1355.35192
[8] Di Castro, A., Kuusi, T. and Palatucci, G.: Nonlocal Harnack inequalities. J. Funct. Anal.267 (2014), no. 6, 1807–1836. · Zbl 1302.35082
[9] Di Nezza, E., Palatucci, G. and Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math.136 (2012), no. 5, 521–573. · Zbl 1252.46023
[10] Iannizzotto, A., Liu, S., Perera, K. and Squassina, M.: Existence results for fractionalp-Laplacian problems via Morse theory. Adv. Calc. Var. 9 (2016), no. 2, 101–125. · Zbl 06567151
[11] Iannizzotto, A., Mosconi, S. and Squassina, M.:HsversusC0-weighted minimizers. NoDEA Nonlinear Differential Equations Appl.22 (2015), no. 3, 477–497. · Zbl 1339.35201
[12] Iannizzotto, A., Mosconi, S. and Squassina, M.:, A note on global regularity for the weak solutions of fractionalp-Laplacian equations. Atti Accad. Naz. Lincei 1392A. Iannizzotto, S. Mosconi and M. Squassina · Zbl 1336.35360
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.