# American Institute of Mathematical Sciences

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. Chen, F. 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

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. Chen, F. 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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