Hölder estimates for viscosity solutions of equations of fractional p-Laplace type

Nonlinear Differential Equations and Applications

Volumn 23, Issue 5, 2016, Pages

  Lindgren, Erik ^a  

^a ROYAL INSTITUTE OF TECHNOLOGY   (Sweden)

Author keywords

35B65; 35D40; 35J60; 35J70; 35R09

Indexed keywords

EID: 84988358251     PISSN: 10219722     EISSN: 14209004     Source Type: Journal    
DOI: 10.1007/s00030-016-0406-x     Document Type: Article

References (35)

1. Moser J.: On Harnack’s theorem for elliptic differential equations. Commun. Pure Appl. Math. 14, 577–591 (1961)
2. Bjorland C., Caffarelli L., Figalli A.: Nonlocal tug-of-war and the infinity fractional Laplacian. Commun. Pure Appl. Math. 65(3), 337–380 (2012)
3. 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)
4. 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)
5. 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)
6. Chasseigne, E., Jakobsen, E.: On nonlocal quasilinear equations and their local limits (2015)
7. Chambolle A., Lindgren E., Monneau R.: A Hölder infinity Laplacian. ESAIM Control Optim. Calc. Var. 18(3), 799–835 (2012)
8. Caffarelli L., Silvestre L.: Regularity theory for fully nonlinear integro-differential equations. Commun. Pure Appl. Math. 62(5), 597–638 (2009)
9. Di Castro A., Kuusi T., Palatucci G.: Nonlocal Harnack inequalities. J. Funct. Anal. 267(6), 1807–1836 (2014)
10. 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)
11. Delarue F.: Krylov and Safonov estimates for degenerate quasilinear elliptic PDEs. J. Differ. Equ. 248(4), 924–951 (2010)
12. 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)
13. 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)
14. 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)
15. 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)
16. 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)
17. 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)
18. Ishii H., Lions P.-L.: Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differ. Equ. 83(1), 26–78 (1990)
19. 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)
20. Iannizzotto, A., Mosconi, S., Squassina, M.: Global Hölder regularity for the fractional p-Laplacian. Rev. Mat. Iberoamer 1–14 (2015) (to appear)
21. 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)
22. 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)
23. α regularity of solutions of some degenerate fully non-linear elliptic equations. Adv. Math. 233, 196–206 (2013)
24. Imbert, C., Silvestre, L.: Estimates on elliptic equations that hold only where the gradient is large. JEMS (2013) (to appear)
25. Kassmann M.: A priori estimates for integro-differential operators with measurable kernels. Calc. Var. Partial Differ. Equ. 34(1), 1–21 (2009)
26. Korvenpää, J., Kuusi, T., Lindgren, E.: Equivalence of solutions to fractional p-Laplace type equations J. Math. Pures Appl. (2016) (to appear)
27. Korvenpää, J., Kuusi, T., Palatucci, G.: Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations (2016)
28. 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)
29. 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)
30. Lindgren, E., Lindqvist, P.: Perron’s method and Wiener’s theorem for a nonlocal equation (2016)
31. Moser J.: On Harnack’s theorem for elliptic differential equations. Commun. Pure Appl. Math. 14, 577–591 (1961)
32. Nash J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)
33. Silvestre L.: Hölder estimates for solutions of integro-differential equations like the fractional Laplace. Indiana Univ. Math. J. 55(3), 1155–1174 (2006)
34. 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)
35. Zhikov V.V.: On Lavrentiev’s phenomenon. Russ. J. Math. Phys. 3(2), 249–269 (1995)