March  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 and 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.  Google Scholar

[2]

C. ChenF. Han and 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.  Google Scholar

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G. De Philippis and 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.  Google Scholar

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

[5]

L. Forzani and 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.  Google Scholar

[6]

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

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

[8]

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

[9]

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

[10]

F. Jiang and 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.  Google Scholar

[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.  Google Scholar

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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.   Google Scholar

[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.  Google Scholar

show all references

References:
[1]

L. Caffarelli and 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.  Google Scholar

[2]

C. ChenF. Han and 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.  Google Scholar

[3]

G. De Philippis and 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.  Google Scholar

[4]

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

[5]

L. Forzani and 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.  Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

F. Jiang and 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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