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doi: 10.3934/cpaa.2021024

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

Arunima Bhattacharya 1, and  Micah Warren 2,,

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.

Keywords: Non-linear, uniformly elliptic, viscosity solutions, Cordes-Nirenberg, non concave/convex.
Mathematics Subject Classification: Primary: 35A09, 35B45; Secondary: 35B20.
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. Armstrong, L. 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. Cao, D. 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

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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. Armstrong, L. 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. Cao, D. 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

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