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

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April 2021, 20(4): 1363-1383. 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 April 2021 Early access 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 and Applied Analysis, 2021, 20 (4) : 1363-1383. 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.
[2]	L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Soc., 1995. doi: 10.1090/coll/043.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[12]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.
[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.
[14]	N. V. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations, Math. USSR-Izv., 20 (1983), 459-492.
[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.
[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.
[17]	N. Nadirashvili, V. Tkachev and S. Vlăduţ, Nonlinear Elliptic Equations and Nonassociative Algebras, American Mathematical Soc., 2014. doi: 10.1090/surv/200.
[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.
[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.
[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.
[21]	V. P. Pingali, $C^{2, \alpha}$estimates and existence results for a nonconcave PDE, Electron. J. Differ. Equ., 168 (2016), 1-10.
[22]	O. Savin, Small perturbation solutions for elliptic equations, Commun Partial Differ. Equ., 3 (2007), 557-578. doi: 10.1080/03605300500394405.
[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.
[24]	G. Talenti, Sopra una classe di equazioni ellittiche a coefficienti misurabili, Ann. Mat. Pura Appl., 69 (1965), 285-304. doi: 10.1007/BF02414375.
[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.

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.
[2]	L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Soc., 1995. doi: 10.1090/coll/043.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[12]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.
[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.
[14]	N. V. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations, Math. USSR-Izv., 20 (1983), 459-492.
[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.
[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.
[17]	N. Nadirashvili, V. Tkachev and S. Vlăduţ, Nonlinear Elliptic Equations and Nonassociative Algebras, American Mathematical Soc., 2014. doi: 10.1090/surv/200.
[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.
[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.
[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.
[21]	V. P. Pingali, $C^{2, \alpha}$estimates and existence results for a nonconcave PDE, Electron. J. Differ. Equ., 168 (2016), 1-10.
[22]	O. Savin, Small perturbation solutions for elliptic equations, Commun Partial Differ. Equ., 3 (2007), 557-578. doi: 10.1080/03605300500394405.
[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.
[24]	G. Talenti, Sopra una classe di equazioni ellittiche a coefficienti misurabili, Ann. Mat. Pura Appl., 69 (1965), 285-304. doi: 10.1007/BF02414375.
[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.

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