doi: 10.3934/cpaa.2021024

$ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations

1. 

Department of Mathematics, University of Washington, Seattle, WA 98195, USA

2. 

Department of Mathematics, University of Oregon, Eugene, OR 97403, USA

* Corresponding author

Received  September 2020 Revised  December 2020 Published  March 2021

In this paper, we show explicit $ C^{2, \alpha} $ interior estimates for viscosity solutions of fully non-linear, uniformly elliptic equations, which are close to linear equations and we also compute an explicit bound for the closeness.

Citation: Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021024
References:
[1]

S. N. ArmstrongL. E. Silvestre and C. K. Smart, Partial regularity of solutions of fully nonlinear, uniformly elliptic equations, Commun. Pure Appl. Math., 65 (2012), 1169-1184.  doi: 10.1002/cpa.21394.  Google Scholar

[2]

L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Soc., 1995. doi: 10.1090/coll/043.  Google Scholar

[3]

L. A. Caffarelli and Y. Yuan, A priori estimates for solutions of fully nonlinear equations with convex level set, Indiana Univ. Math. J., 49 (2000), 681-695.  doi: 10.1512/iumj.2000.49.1901.  Google Scholar

[4]

X. Cabré and L. A. Caffarelli, Interior $C^{2, \alpha}$ regularity theory for a class of nonconvex fully nonlinear elliptic equations, J. Math. Pures Appl., 82 (2003), 573-612.  doi: 10.1016/S0021-7824(03)00029-1.  Google Scholar

[5]

E. Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens, Michigan Math. J., 5 (1958), 105-126.   Google Scholar

[6]

Y. CaoD. S. Li and L. H. Wang, A priori estimates for classical solutions of fully nonlinear elliptic equations, Sci. China Math., 54 (2011), 457-462.  doi: 10.1007/s11425-010-4092-6.  Google Scholar

[7]

T. C. Collins, $C^{2, \alpha}$ estimates for nonlinear elliptic equations of twisted type, Calc. Var. Partial Differ. Equ., 55 (2016), 1-11.  doi: 10.1007/s00526-015-0950-y.  Google Scholar

[8]

H. O. Cordes, Über die erste randwertaufgabe bei quasilinearen differentialgleichungen zweiter ordnung in mehr als zwei variablen, Math. Ann., 131 (1956), 278-312.  doi: 10.1007/BF01342965.  Google Scholar

[9]

H. O. Cordes, Zero order a priori estimates for solutions of elliptic differential equations, in Proc. Sympos. Pure Math., American Mathematical Society, Providence, R.I., 1961.  Google Scholar

[10]

H. J. Dong, Recent progress in the $ l\_p $ theory for elliptic and parabolic equations with discontinuous coefficients, Anal. Theor. Appl., 36 (2020), 161-199.  doi: 10.4208/ata.oa-0021.  Google Scholar

[11]

L. C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Commun. Pure Appl. Math., 35 (1982), 333-363.  doi: 10.1002/cpa.3160350303.  Google Scholar

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.  Google Scholar

[13]

Q. B. Huang, Regularity theory for ln-viscosity solutions to fully nonlinear elliptic equations with asymptotical approximate convexity, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 36 (2019), 1869-1902.  doi: 10.1016/j.anihpc.2019.06.001.  Google Scholar

[14]

N. V. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations, Math. USSR-Izv., 20 (1983), 459-492.   Google Scholar

[15]

N. V. Krylov and M. V. Safonov, A certain property of solutions of parabolic equations with measurable coefficients, Math. USSR-Izv., 16 (1981), 151-164.   Google Scholar

[16]

N. Nadirashvili and S. Vlăduţ, Singular viscosity solutions to fully nonlinear elliptic equations, J. Math. Pures Appl., 89 (2008), 107-113.  doi: 10.1016/j.matpur.2007.10.004.  Google Scholar

[17]

N. Nadirashvili, V. Tkachev and S. Vlăduţ, Nonlinear Elliptic Equations and Nonassociative Algebras, American Mathematical Soc., 2014. doi: 10.1090/surv/200.  Google Scholar

[18]

N. Nadirashvili and S. Vlăduţ, Singular solution to special lagrangian equations, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 27 (2010), 1179-1188.  doi: 10.1016/j.anihpc.2010.05.001.  Google Scholar

[19]

L. Nirenberg, On nonlinear elliptic partial differential equations and Hölder continuity, Commun. Pure Appl. Math., 6 (1953), 103-156.  doi: 10.1002/cpa.3160060105.  Google Scholar

[20]

L. Nirenberg, On a generalization of quasi-conformal mappings and its application to elliptic partial differential equations, in Contributions to the Theory of Partial Differential Equations, Princeton University Press, Princeton, N. J., 1954.  Google Scholar

[21]

V. P. Pingali, $C^{2, \alpha}$estimates and existence results for a nonconcave PDE, Electron. J. Differ. Equ., 168 (2016), 1-10.   Google Scholar

[22]

O. Savin, Small perturbation solutions for elliptic equations, Commun Partial Differ. Equ., 3 (2007), 557-578.  doi: 10.1080/03605300500394405.  Google Scholar

[23]

J. Streets and M. Warren, Evans-Krylov estimates for a nonconvex Monge-Ampère equation, Math. Ann., 365 (2016), 805-834.  doi: 10.1007/s00208-015-1293-x.  Google Scholar

[24]

G. Talenti, Sopra una classe di equazioni ellittiche a coefficienti misurabili, Ann. Mat. Pura Appl., 69 (1965), 285-304.  doi: 10.1007/BF02414375.  Google Scholar

[25]

Y. Yuan, A priori estimates for solutions of fully nonlinear special Lagrangian equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 261-270.  doi: 10.1016/S0294-1449(00)00065-2.  Google Scholar

show all references

References:
[1]

S. N. ArmstrongL. E. Silvestre and C. K. Smart, Partial regularity of solutions of fully nonlinear, uniformly elliptic equations, Commun. Pure Appl. Math., 65 (2012), 1169-1184.  doi: 10.1002/cpa.21394.  Google Scholar

[2]

L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Soc., 1995. doi: 10.1090/coll/043.  Google Scholar

[3]

L. A. Caffarelli and Y. Yuan, A priori estimates for solutions of fully nonlinear equations with convex level set, Indiana Univ. Math. J., 49 (2000), 681-695.  doi: 10.1512/iumj.2000.49.1901.  Google Scholar

[4]

X. Cabré and L. A. Caffarelli, Interior $C^{2, \alpha}$ regularity theory for a class of nonconvex fully nonlinear elliptic equations, J. Math. Pures Appl., 82 (2003), 573-612.  doi: 10.1016/S0021-7824(03)00029-1.  Google Scholar

[5]

E. Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens, Michigan Math. J., 5 (1958), 105-126.   Google Scholar

[6]

Y. CaoD. S. Li and L. H. Wang, A priori estimates for classical solutions of fully nonlinear elliptic equations, Sci. China Math., 54 (2011), 457-462.  doi: 10.1007/s11425-010-4092-6.  Google Scholar

[7]

T. C. Collins, $C^{2, \alpha}$ estimates for nonlinear elliptic equations of twisted type, Calc. Var. Partial Differ. Equ., 55 (2016), 1-11.  doi: 10.1007/s00526-015-0950-y.  Google Scholar

[8]

H. O. Cordes, Über die erste randwertaufgabe bei quasilinearen differentialgleichungen zweiter ordnung in mehr als zwei variablen, Math. Ann., 131 (1956), 278-312.  doi: 10.1007/BF01342965.  Google Scholar

[9]

H. O. Cordes, Zero order a priori estimates for solutions of elliptic differential equations, in Proc. Sympos. Pure Math., American Mathematical Society, Providence, R.I., 1961.  Google Scholar

[10]

H. J. Dong, Recent progress in the $ l\_p $ theory for elliptic and parabolic equations with discontinuous coefficients, Anal. Theor. Appl., 36 (2020), 161-199.  doi: 10.4208/ata.oa-0021.  Google Scholar

[11]

L. C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Commun. Pure Appl. Math., 35 (1982), 333-363.  doi: 10.1002/cpa.3160350303.  Google Scholar

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.  Google Scholar

[13]

Q. B. Huang, Regularity theory for ln-viscosity solutions to fully nonlinear elliptic equations with asymptotical approximate convexity, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 36 (2019), 1869-1902.  doi: 10.1016/j.anihpc.2019.06.001.  Google Scholar

[14]

N. V. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations, Math. USSR-Izv., 20 (1983), 459-492.   Google Scholar

[15]

N. V. Krylov and M. V. Safonov, A certain property of solutions of parabolic equations with measurable coefficients, Math. USSR-Izv., 16 (1981), 151-164.   Google Scholar

[16]

N. Nadirashvili and S. Vlăduţ, Singular viscosity solutions to fully nonlinear elliptic equations, J. Math. Pures Appl., 89 (2008), 107-113.  doi: 10.1016/j.matpur.2007.10.004.  Google Scholar

[17]

N. Nadirashvili, V. Tkachev and S. Vlăduţ, Nonlinear Elliptic Equations and Nonassociative Algebras, American Mathematical Soc., 2014. doi: 10.1090/surv/200.  Google Scholar

[18]

N. Nadirashvili and S. Vlăduţ, Singular solution to special lagrangian equations, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 27 (2010), 1179-1188.  doi: 10.1016/j.anihpc.2010.05.001.  Google Scholar

[19]

L. Nirenberg, On nonlinear elliptic partial differential equations and Hölder continuity, Commun. Pure Appl. Math., 6 (1953), 103-156.  doi: 10.1002/cpa.3160060105.  Google Scholar

[20]

L. Nirenberg, On a generalization of quasi-conformal mappings and its application to elliptic partial differential equations, in Contributions to the Theory of Partial Differential Equations, Princeton University Press, Princeton, N. J., 1954.  Google Scholar

[21]

V. P. Pingali, $C^{2, \alpha}$estimates and existence results for a nonconcave PDE, Electron. J. Differ. Equ., 168 (2016), 1-10.   Google Scholar

[22]

O. Savin, Small perturbation solutions for elliptic equations, Commun Partial Differ. Equ., 3 (2007), 557-578.  doi: 10.1080/03605300500394405.  Google Scholar

[23]

J. Streets and M. Warren, Evans-Krylov estimates for a nonconvex Monge-Ampère equation, Math. Ann., 365 (2016), 805-834.  doi: 10.1007/s00208-015-1293-x.  Google Scholar

[24]

G. Talenti, Sopra una classe di equazioni ellittiche a coefficienti misurabili, Ann. Mat. Pura Appl., 69 (1965), 285-304.  doi: 10.1007/BF02414375.  Google Scholar

[25]

Y. Yuan, A priori estimates for solutions of fully nonlinear special Lagrangian equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 261-270.  doi: 10.1016/S0294-1449(00)00065-2.  Google Scholar

[1]

Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $ BV $ solution for a non-linear Hamilton-Jacobi system. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3273-3293. doi: 10.3934/dcds.2020405

[2]

Suzete Maria Afonso, Vanessa Ramos, Jaqueline Siqueira. Equilibrium states for non-uniformly hyperbolic systems: Statistical properties and analyticity. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021045

[3]

Craig Cowan. Supercritical elliptic problems involving a Cordes like operator. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021037

[4]

Pengyu Chen. Periodic solutions to non-autonomous evolution equations with multi-delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2921-2939. doi: 10.3934/dcdsb.2020211

[5]

Shoichi Hasegawa, Norihisa Ikoma, Tatsuki Kawakami. On weak solutions to a fractional Hardy–Hénon equation: Part I: Nonexistence. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021033

[6]

Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055

[7]

Kazeem Olalekan Aremu, Chinedu Izuchukwu, Grace Nnenanya Ogwo, Oluwatosin Temitope Mewomo. Multi-step iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2161-2180. doi: 10.3934/jimo.2020063

[8]

Alexander Tolstonogov. BV solutions of a convex sweeping process with a composed perturbation. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021012

[9]

Lipeng Duan, Jun Yang. On the non-degeneracy of radial vortex solutions for a coupled Ginzburg-Landau system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021056

[10]

Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329

[11]

Beixiang Fang, Qin Zhao. Uniqueness of steady 1-D shock solutions in a finite nozzle via vanishing viscosity aguments. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021066

[12]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087

[13]

Pascal Noble, Sebastien Travadel. Non-persistence of roll-waves under viscous perturbations. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 61-70. doi: 10.3934/dcdsb.2001.1.61

[14]

Vieri Benci, Marco Cococcioni. The algorithmic numbers in non-archimedean numerical computing environments. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1673-1692. doi: 10.3934/dcdss.2020449

[15]

Liqin Qian, Xiwang Cao. Character sums over a non-chain ring and their applications. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020134

[16]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2677-2698. doi: 10.3934/dcds.2020381

[17]

Cheng-Kai Hu, Fung-Bao Liu, Hong-Ming Chen, Cheng-Feng Hu. Network data envelopment analysis with fuzzy non-discretionary factors. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1795-1807. doi: 10.3934/jimo.2020046

[18]

Siting Liu, Levon Nurbekyan. Splitting methods for a class of non-potential mean field games. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021014

[19]

Antonio De Rosa, Domenico Angelo La Manna. A non local approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021059

[20]

Joel Coacalle, Andrew Raich. Compactness of the complex Green operator on non-pseudoconvex CR manifolds. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021061

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (17)
  • HTML views (76)
  • Cited by (0)

Other articles
by authors

[Back to Top]