2018, 38(3): 1427-1440. doi: 10.3934/dcds.2018058

On interior $C^2$-estimates for the Monge-Ampère equation

Department of Mathematics, Kansas State University, 138 Cardwell Hall, Manhattan, KS 66506-0903, USA

Received  May 2017 Revised  October 2017 Published  December 2017

Fund Project: The author is supported by NSF grant DMS 1361754.

An approach towards apriori interior $C^2$-estimates for the Monge-Ampère equation based on a mean-value inequality for nonnegative subsolutions to the linearized Monge-Ampère equation is implemented.

Citation: Diego Maldonado. On interior $C^2$-estimates for the Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1427-1440. doi: 10.3934/dcds.2018058
References:
[1]

L. Caffarelli, C. Gutiérrez, Properties of the solutions of the linearized Monge-Ampère equation, Amer. J. Math., 119 (1997), 423-465. doi: 10.1353/ajm.1997.0010.

[2]

C. Chen, F. Han, Q. Ou, The interior $C^2$ estimate for the Monge-Ampère equation in dimension $n=2$, Analysis PDE., 9 (2016), 1419-1432. doi: 10.2140/apde.2016.9.1419.

[3]

G. De Philippis, A. Figalli, The Monge-Ampère equation and its link to optimal transportation, Bull. Amer. Math. Soc., 51 (2014), 527-580. doi: 10.1090/S0273-0979-2014-01459-4.

[4]

A. Figalli, The Monge-Ampère Equation and Its Applications Zurich Lectures in Advanced Mathematics. European Mathematical Society, 2017.

[5]

L. Forzani, D. Maldonado, Properties of the solutions to the Monge-Ampère equation, Nonlinear Anal., 57 (2004), 815-829. doi: 10.1016/j.na.2004.03.019.

[6]

D. Gilbarg and M. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Verlag, 2001.

[7]

D. Gilbarg and M. Trudinger, Elliptic Partial Differential Equations of Second Order Springer Verlag, 2001.

[8]

C. Gutiérrez, The Monge-Ampère Equation Progress in Nonlinear Differential Equations and Their Applications, volume 44. Birkäuser, 2001.

[9]

Q. Han and F. -H. Lin, Elliptic Partial Differential Equations Courant Lecture Notes, vol. 1. American Mathematical Society, Providence, RI, 2011.

[10]

F. Jiang, N. Trudinger, On Pogorelov estimates in optimal transportation and geometric optics, Bull. Math. Sci., 4 (2014), 407-431. doi: 10.1007/s13373-014-0055-5.

[11]

D. Maldonado, Harnack's inequality for solutions to the linearized Monge-Ampère operator with lower-order terms, J. Differential Equations, 256 (2014), 1987-2022. doi: 10.1016/j.jde.2013.12.013.

[12]

A. V. Pogorelov, The regularity of the generalized solutions of the equation $\det (\partial^2u/\partial x^i \partial x^j) = \varphi(x^1, x^2, ···, x^n) >0$ (Russian), Dokl. Akad. Nauk SSSR, 200 (1971), 534-537.

[13]

N. Trudinger and X.-J. Wang, The Monge-Ampère equation and its geometric applications, in Handbook of Geometric Analysis. No. 1 (eds. L. Ji, P. Li, R. Schoen and L. Simon), Adv. Lect. Math. (ALM), International Press of Boston, 7 (2008), 467-524.

show all references

References:
[1]

L. Caffarelli, C. Gutiérrez, Properties of the solutions of the linearized Monge-Ampère equation, Amer. J. Math., 119 (1997), 423-465. doi: 10.1353/ajm.1997.0010.

[2]

C. Chen, F. Han, Q. Ou, The interior $C^2$ estimate for the Monge-Ampère equation in dimension $n=2$, Analysis PDE., 9 (2016), 1419-1432. doi: 10.2140/apde.2016.9.1419.

[3]

G. De Philippis, A. Figalli, The Monge-Ampère equation and its link to optimal transportation, Bull. Amer. Math. Soc., 51 (2014), 527-580. doi: 10.1090/S0273-0979-2014-01459-4.

[4]

A. Figalli, The Monge-Ampère Equation and Its Applications Zurich Lectures in Advanced Mathematics. European Mathematical Society, 2017.

[5]

L. Forzani, D. Maldonado, Properties of the solutions to the Monge-Ampère equation, Nonlinear Anal., 57 (2004), 815-829. doi: 10.1016/j.na.2004.03.019.

[6]

D. Gilbarg and M. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Verlag, 2001.

[7]

D. Gilbarg and M. Trudinger, Elliptic Partial Differential Equations of Second Order Springer Verlag, 2001.

[8]

C. Gutiérrez, The Monge-Ampère Equation Progress in Nonlinear Differential Equations and Their Applications, volume 44. Birkäuser, 2001.

[9]

Q. Han and F. -H. Lin, Elliptic Partial Differential Equations Courant Lecture Notes, vol. 1. American Mathematical Society, Providence, RI, 2011.

[10]

F. Jiang, N. Trudinger, On Pogorelov estimates in optimal transportation and geometric optics, Bull. Math. Sci., 4 (2014), 407-431. doi: 10.1007/s13373-014-0055-5.

[11]

D. Maldonado, Harnack's inequality for solutions to the linearized Monge-Ampère operator with lower-order terms, J. Differential Equations, 256 (2014), 1987-2022. doi: 10.1016/j.jde.2013.12.013.

[12]

A. V. Pogorelov, The regularity of the generalized solutions of the equation $\det (\partial^2u/\partial x^i \partial x^j) = \varphi(x^1, x^2, ···, x^n) >0$ (Russian), Dokl. Akad. Nauk SSSR, 200 (1971), 534-537.

[13]

N. Trudinger and X.-J. Wang, The Monge-Ampère equation and its geometric applications, in Handbook of Geometric Analysis. No. 1 (eds. L. Ji, P. Li, R. Schoen and L. Simon), Adv. Lect. Math. (ALM), International Press of Boston, 7 (2008), 467-524.

[1]

Qi-Rui Li, Xu-Jia Wang. Regularity of the homogeneous Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6069-6084. doi: 10.3934/dcds.2015.35.6069

[2]

Alessio Figalli, Young-Heon Kim. Partial regularity of Brenier solutions of the Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 559-565. doi: 10.3934/dcds.2010.28.559

[3]

Bo Guan, Qun Li. A Monge-Ampère type fully nonlinear equation on Hermitian manifolds. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1991-1999. doi: 10.3934/dcdsb.2012.17.1991

[4]

Adam M. Oberman. Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 221-238. doi: 10.3934/dcdsb.2008.10.221

[5]

Jiakun Liu, Neil S. Trudinger. On Pogorelov estimates for Monge-Ampère type equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1121-1135. doi: 10.3934/dcds.2010.28.1121

[6]

Barbara Brandolini, Carlo Nitsch, Cristina Trombetti. Shape optimization for Monge-Ampère equations via domain derivative. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 825-831. doi: 10.3934/dcdss.2011.4.825

[7]

Limei Dai. Multi-valued solutions to a class of parabolic Monge-Ampère equations. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1061-1074. doi: 10.3934/cpaa.2014.13.1061

[8]

Jingang Xiong, Jiguang Bao. The obstacle problem for Monge-Ampère type equations in non-convex domains. Communications on Pure & Applied Analysis, 2011, 10 (1) : 59-68. doi: 10.3934/cpaa.2011.10.59

[9]

Cristian Enache. Maximum and minimum principles for a class of Monge-Ampère equations in the plane, with applications to surfaces of constant Gauss curvature. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1347-1359. doi: 10.3934/cpaa.2014.13.1347

[10]

Shouchuan Hu, Haiyan Wang. Convex solutions of boundary value problem arising from Monge-Ampère equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 705-720. doi: 10.3934/dcds.2006.16.705

[11]

Haitao Yang, Yibin Chang. On the blow-up boundary solutions of the Monge -Ampére equation with singular weights. Communications on Pure & Applied Analysis, 2012, 11 (2) : 697-708. doi: 10.3934/cpaa.2012.11.697

[12]

Gregorio Díaz, Jesús Ildefonso Díaz. On the free boundary associated with the stationary Monge--Ampère operator on the set of non strictly convex functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1447-1468. doi: 10.3934/dcds.2015.35.1447

[13]

Ovidiu Savin. A Liouville theorem for solutions to the linearized Monge-Ampere equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 865-873. doi: 10.3934/dcds.2010.28.865

[14]

Ciro D'Apice, Peter I. Kogut, Rosanna Manzo. On optimization of a highly re-entrant production system. Networks & Heterogeneous Media, 2016, 11 (3) : 415-445. doi: 10.3934/nhm.2016003

[15]

Jesus Garcia Azorero, Juan J. Manfredi, I. Peral, Julio D. Rossi. Limits for Monge-Kantorovich mass transport problems. Communications on Pure & Applied Analysis, 2008, 7 (4) : 853-865. doi: 10.3934/cpaa.2008.7.853

[16]

Zuo Quan Xu, Jia-An Yan. A note on the Monge-Kantorovich problem in the plane. Communications on Pure & Applied Analysis, 2015, 14 (2) : 517-525. doi: 10.3934/cpaa.2015.14.517

[17]

Hal L. Smith. Monotone dynamical systems: Reflections on new advances & applications. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 485-504. doi: 10.3934/dcds.2017020

[18]

Jean-Michel Coron, Matthias Kawski, Zhiqiang Wang. Analysis of a conservation law modeling a highly re-entrant manufacturing system. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1337-1359. doi: 10.3934/dcdsb.2010.14.1337

[19]

Anthony W. Leung, Xiaojie Hou, Wei Feng. Traveling wave solutions for Lotka-Volterra system re-visited. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 171-196. doi: 10.3934/dcdsb.2011.15.171

[20]

Abbas Moameni. Invariance properties of the Monge-Kantorovich mass transport problem. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2653-2671. doi: 10.3934/dcds.2016.36.2653

2016 Impact Factor: 1.099

Metrics

  • PDF downloads (9)
  • HTML views (19)
  • Cited by (0)

Other articles
by authors

[Back to Top]