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On problems with weighted elliptic operator and general growth nonlinearities
$ 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 |
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.
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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