Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-22T22:40:11.707Z Has data issue: false hasContentIssue false
  • English
  • Français

Regularity for solutions of fully nonlinear elliptic equations with nonhomogeneous degeneracy

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  29 January 2020

Cristiana De Filippis
Cristiana De Filippis*
Affiliation:
Cristiana De Filippis Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX26GG, United Kingdom ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that viscosity solutions to fully nonlinear elliptic equations with degeneracy of double phase type are locally C1, γ-regular.

Keywords

Fully nonlinear degenerate equationsDouble phase problemsRegularity

MSC classification

Primary: 35J60: Nonlinear elliptic equations
Secondary: 35J70: Degenerate elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 1 , February 2021 , pp. 110 - 132
Check for updates
Copyright
Copyright © The Author(s) 2020. Published by The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Arroyo, A., Heino, J. and Parviainen, M.. Tug-of-war games with varying probabilities and the normalized p(x)-Laplacian. Commun. Pure Appl. Anal. 16 (2017), 915–944.CrossRefGoogle Scholar
2Attouchi, A., Parviainen, M. and Ruosteenoja, E.. C 1, α-regularity for the normalized p-Poisson problem. J. Math. Pures. Appl. 108 (2017), 553–591.CrossRefGoogle Scholar
3Banerjee, A. and Munive, I. H., Gradient continuity estimates for normalized p-Poisson equation. Comm. Cont. Math., to appear.Google Scholar
4Barles, G. and Imbert, C.. Second-order elliptic integro-differential equations: viscosity solutions' theory revisited. Ann. I. H. Poincaré - AN 25 (2008), 567–585.CrossRefGoogle Scholar
5Baroni, P., Colombo, M. and Mingione, G.. Harnack inequalities for double phase functionals. Nonlinear Anal. 121 (2015), 206–222.CrossRefGoogle Scholar
6Baroni, P., Colombo, M. and Mingione, G.. Regularity for general functionals with double phase. Calc. Var. PDE 57 (2018), 62.CrossRefGoogle Scholar
7Birindelli, I. and Demengel, F.. Comparison principle and Liouville type results for singular fully nonlinear operators. Ann. Fac. Sci. Toulouse Math. 13 (2004), 261–287.CrossRefGoogle Scholar
8Birindelli, I. and Demengel, F.. Regularity and uniqueness of the first eigenfunction for singular, fully nonlinear elliptic operators. J. Differ. Equ. 249 (2010), 1089–1110.CrossRefGoogle Scholar
9Birindelli, I. and Demengel, F.. Eigenvalue, maximum principle and regularity for fully nonlinear homogeneous operators. Comm. Pure Appl. Anal. 6 (2007), 335–366.CrossRefGoogle Scholar
10Birindelli, I. and Demengel, F.. Regularity for radial solutions of degenerate fully nonlinear equations. Nonlinear Anal. 75 (2012), 6237–6249.CrossRefGoogle Scholar
11Birindelli, I. and Demengel, F.. C 1, β regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations. ESAIM Control Optim. Calc. Var. 20 (2014), 1009–1024.CrossRefGoogle Scholar
12Bronzi, A. C., Pimentel, E. A., Rampasso, G. C. and Teixeira, E. V., Regularity of solutions to a class of variable-exponent fully nonlinear elliptic equations. Preprint (2018). https://arxiv.org/abs/1812.11428.Google Scholar
13Caffarelli, L. and Cabré, X., 1995 Fully Nonlinear Elliptic Equations. American Mathematical Society, 43. Providence, RI: AMS Colloquium Publications.Google Scholar
14Chlebicka, I. and De Filippis, C., Removable sets in non-uniformly elliptic problems. Ann. Mat. Pura Appl. https://doi.org/10.1007/s10231-019-00894-1.CrossRefGoogle Scholar
15Colombo, M. and Mingione, G.. Bounded minimisers of double phase variationals integrals. Arch. Rational Mech. Anal. 218 (2015), 219–273.CrossRefGoogle Scholar
16Colombo, M. and Mingione, G.. Calderón-Zygmund estimates and non-uniformly elliptic operators. J. Funct. Anal. 270 (2016), 1416–1478.CrossRefGoogle Scholar
17Colombo, M. and Mingione, G.. Regularity for double phase variational problems. Arch. Rational Mech. Anal. 215 (2015), 443–496.CrossRefGoogle Scholar
18Crandall. H. Ishii, M. and Lions, P.-L.. User's guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27 (1992), 1–67.CrossRefGoogle Scholar
19Cruz-Uribe, D. and Hästo, P.. Extrapolation and interpolation in generalized Orlicz spaces. Trans. Amer. Math. Soc. 370 (2018), 4323–4349.CrossRefGoogle Scholar
20Dávila, G., Felmer, P. and Quaas, A.. Alexandroff-Bakelman-Pucci estimate for singular or degenerate fully nonlinear elliptic equations. C. R. Math. Acad. Sci. Paris 347 (2009), 1165–1168.CrossRefGoogle Scholar
21Dávila, G., Felmer, P. and Quaas, A.. Harnack inequality for singular fully nonlinear operators and some existence results. Calc. Var. PDE 39 (2010), 557–578.CrossRefGoogle Scholar
22De Filippis, C. and Mingione, G., A borderline case of Calderón-Zygmund estimates for non-uniformly elliptic problems. St Petersburg Math. J., to appear.Google Scholar
23De Filippis, C. and Mingione, G., Manifold constrained non-uniformly elliptic problems. J. Geom. Anal., to appear.Google Scholar
24De Filippis, C. and Palatucci, G.. Hölder regularity for nonlocal double phase equations. J. Differ. Equ. 267 (2019), 547–586.CrossRefGoogle Scholar
25Esposito, L., Leonetti, F. and Mingione, G.. Sharp regularity for functionals with (p, q) growth. J. Differ. Equ. 204 (2004), 5–55.CrossRefGoogle Scholar
26Fonseca, I., Malý, J. and Mingione, G.. Scalar minimizers with fractal singular sets. Arch. Ration. Mech. Anal. 172 (2004), 295–307.CrossRefGoogle Scholar
27Harjulehto, P. and Hästo, P., Double phase image restoration. J. Math. Anal. Appl., to appear.Google Scholar
28Hästo, P.. The maximal operator on generalized Orlicz spaces. J. Funct. Anal. 269 (2015), 4038–4048.CrossRefGoogle Scholar
29Imbert, C.. Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations. J. Differ. Equ. 250 (2011), 1553–1574.CrossRefGoogle Scholar
30Imbert, C. and Silvestre, L.. C 1, α regularity of solutions of some degenerate, fully non-linear elliptic equations. Adv. Math. 233 (2013), 196–206.CrossRefGoogle Scholar
31Lindgren, E.. Hölder estimates for viscosity solutions of equations of fractional p-Laplace type. NoDEA 22 (2016), 55.CrossRefGoogle Scholar
32Marcellini, P.. Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions. Arch. Rat. Mech. Anal. 105 (1989), 267–284.CrossRefGoogle Scholar
33Marcellini, P.. On the definition and the lower semicontinuity of certain quasiconvex integrals. Ann. de l'I.H.P. Anal. non linéaire 3 (1986), 391–409.Google Scholar
34Siltakoski, J.. Equivalence of viscosity and weak solutions for the normalized p(x)-Laplacian. Calc. Var. PDE 57 (2018), 95.CrossRefGoogle Scholar
35Zhikov, V. V.. Lavrentiev phenomenon and homogeneization of some variational problems. C. R. Acad. Sci. Paris Sér I Math. 316 (1993), 435–439.Google Scholar
36Zhikov, V. V.. On Lavrentiev phenomenon. Russian J. Math. Phys. 3 (1995), 249–269.Google Scholar
37Zhikov, V. V.. On some variational problems. Russian J. Math. Phys. 5 (1997), 105–116.Google Scholar