Abstract
We prove Hölder estimates for viscosity solutions of a class of possibly degenerate and singular equations modelled by the fractional p-Laplace equation
where \({s \in (0,1)}\) and \({p > 2}\) or \({1/(1-s) < p < 2}\). Our results also apply for inhomogeneous equations with more general kernels, when p and s are allowed to vary with x, without any regularity assumption on p and s. This complements and extends some of the recently obtained Hölder estimates for weak solutions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Moser J.: On Harnack’s theorem for elliptic differential equations. Commun. Pure Appl. Math. 14, 577–591 (1961)
Bjorland C., Caffarelli L., Figalli A.: Nonlocal tug-of-war and the infinity fractional Laplacian. Commun. Pure Appl. Math. 65(3), 337–380 (2012)
Barles G., Chasseigne E., Imbert C.: Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations. J. Eur. Math. Soc. (JEMS) 13(1), 1–26 (2011)
Birindelli I., Demengel F.: Comparison principle and Liouville type results for singular fully nonlinear operators. Ann. Fac. Sci. Toulouse Math. (6) 13(2), 261–287 (2004)
Birindelli I., Demengel F.: Eigenvalue and Dirichlet problem for fully-nonlinear operators in non-smooth domains. J. Math. Anal. Appl. 352(2), 822–835 (2009)
Chasseigne, E., Jakobsen, E.: On nonlocal quasilinear equations and their local limits (2015)
Chambolle A., Lindgren E., Monneau R.: A Hölder infinity Laplacian. ESAIM Control Optim. Calc. Var. 18(3), 799–835 (2012)
Caffarelli L., Silvestre L.: Regularity theory for fully nonlinear integro-differential equations. Commun. Pure Appl. Math. 62(5), 597–638 (2009)
Di Castro A., Kuusi T., Palatucci G.: Nonlocal Harnack inequalities. J. Funct. Anal. 267(6), 1807–1836 (2014)
Di Castro, A., Kuusi, T., Palatucci, G.: Local behavior of fractional p-minimizers. Ann. Inst. H. Poincaré Anal. Non Linéaire 33(5), 1279–1299 (2016)
Delarue F.: Krylov and Safonov estimates for degenerate quasilinear elliptic PDEs. J. Differ. Equ. 248(4), 924–951 (2010)
Dávila G., Felmer P., Quaas A.: Alexandroff–Bakelman–Pucci estimate for singular or degenerate fully nonlinear elliptic equations. C. R. Math. Acad. Sci. Paris 347(19–20), 1165–1168 (2009)
Dávila G., Felmer P., Quaas A.: Harnack inequality for singular fully nonlinear operators and some existence results. Calc. Var. Partial Differ. Equ. 39(3–4), 557–578 (2010)
De Giorgi E.: Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3, 25–43 (1957)
Elmoataz A., Desquesnes X., Lezoray O.: Non-local morphological pdes and -laplacian equation on graphs with applications in image processing and machine learning. IEEE J. Select. Topics Signal Process. 6(7), 764–779 (2012)
Elmoataz, A., Desquesnes, X., Lakhdari, Z., Olivier L.: Nonlocal infinity laplacian equation on graphs with applications in image processing and machine learning. Math. Comput. Simul. 102,153-163 (2014)
Hästö, P.A.: Counter-examples of regularity in variable exponent Sobolev spaces. The p-harmonic equation and recent advances in analysis. Contemp. Math., vol. 370, pp. 133–143. Amer. Math. Soc., Providence (2005)
Ishii H., Lions P.-L.: Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differ. Equ. 83(1), 26–78 (1990)
Imbert C.: Alexandroff–Bakelman–Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations. J. Differ. Equ. 250(3), 1553–1574 (2011)
Iannizzotto, A., Mosconi, S., Squassina, M.: Global Hölder regularity for the fractional p-Laplacian. Rev. Mat. Iberoamer 1–14 (2015) (to appear)
Iannizzotto A., Mosconi S., Squassina M.: A note on global regularity for the weak solutions of fractional. p-Laplacian equations. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27(1), 15–24 (2016)
Ishii H., Nakamura G.: A class of integral equations and approximation of p-Laplace equations. Calc. Var. Partial Differ. Equ. 37(3–4), 485–522 (2010)
Imbert C., Silvestre L.: \({C^{1,\alpha}}\) regularity of solutions of some degenerate fully non-linear elliptic equations. Adv. Math. 233, 196–206 (2013)
Imbert, C., Silvestre, L.: Estimates on elliptic equations that hold only where the gradient is large. JEMS (2013) (to appear)
Kassmann M.: A priori estimates for integro-differential operators with measurable kernels. Calc. Var. Partial Differ. Equ. 34(1), 1–21 (2009)
Korvenpää, J., Kuusi, T., Lindgren, E.: Equivalence of solutions to fractional p-Laplace type equations J. Math. Pures Appl. (2016) (to appear)
Korvenpää, J., Kuusi, T., Palatucci, G.: Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations (2016)
Kuusi T., Mingione G., Sire Y.: A fractional Gehring lemma, with applications to nonlocal equations. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 25(4), 345–358 (2014)
Krylov N.V., Safonov M.V.: An estimate for the probability of a diffusion process hitting a set of positive measure. Dokl. Akad. Nauk SSSR 245(1), 18–20 (1979)
Lindgren, E., Lindqvist, P.: Perron’s method and Wiener’s theorem for a nonlocal equation (2016)
Moser J.: On Harnack’s theorem for elliptic differential equations. Commun. Pure Appl. Math. 14, 577–591 (1961)
Nash J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)
Silvestre L.: Hölder estimates for solutions of integro-differential equations like the fractional Laplace. Indiana Univ. Math. J. 55(3), 1155–1174 (2006)
Zhikov, V.V.: Averaging of functionals of the calculus of variations and elasticity theory. Izv. Akad. Nauk SSSR Ser. Mat. 50(4), 675–710, 877 (1986)
Zhikov V.V.: On Lavrentiev’s phenomenon. Russ. J. Math. Phys. 3(2), 249–269 (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the Swedish Research Council, Grant No. 2012-3124. Partially supported by the Royal Swedish Academy of Sciences. A substantial part of this work was carried out at the Isaac Newton’s Institute, during the program “Free boundary problems” and I am thankful for the great hospitality.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Lindgren, E. Hölder estimates for viscosity solutions of equations of fractional p-Laplace type. Nonlinear Differ. Equ. Appl. 23, 55 (2016). https://doi.org/10.1007/s00030-016-0406-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00030-016-0406-x