# American Institute of Mathematical Sciences

April  2015, 35(4): 1447-1468. doi: 10.3934/dcds.2015.35.1447

## On the free boundary associated with the stationary Monge--Ampère operator on the set of non strictly convex functions

Received  June 2013 Revised  October 2013 Published  November 2014

This paper deals with several qualitative properties of solutions of some stationary equations associated to the Monge--Ampère operator on the set of convex functions which are not necessarily understood in a strict sense. Mainly, we focus our attention on the occurrence of a free boundary (separating the region where the solution $u$ is locally a hyperplane, thus, the Hessian $D^{2}u$ is vanishing, from the rest of the domain). In particular, our results apply to suitable formulations of the Gauss curvature flow and of the worn stones problems intensively studied in the literature.
Citation: 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
##### References:

show all references

##### References:
 [1] 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 [2] 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 [3] 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 [4] 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 [5] 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 [6] Luca Codenotti, Marta Lewicka. Visualization of the convex integration solutions to the Monge-Ampère equation. Evolution Equations & Control Theory, 2019, 8 (2) : 273-300. doi: 10.3934/eect.2019015 [7] 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 [8] Roberto Paroni, Podio-Guidugli Paolo, Brian Seguin. On the nonlocal curvatures of surfaces with or without boundary. Communications on Pure & Applied Analysis, 2018, 17 (2) : 709-727. doi: 10.3934/cpaa.2018037 [9] 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 [10] 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 [11] 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 [12] Zhijun Zhang. Optimal global asymptotic behavior of the solution to a singular monge-ampère equation. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1129-1145. doi: 10.3934/cpaa.2020053 [13] 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 [14] 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 [15] Fan Cui, Huaiyu Jian. Symmetry of solutions to a class of Monge-Ampère equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1247-1259. doi: 10.3934/cpaa.2019060 [16] Limei Dai, Hongyu Li. Entire subsolutions of Monge-Ampère type equations. Communications on Pure & Applied Analysis, 2020, 19 (1) : 19-30. doi: 10.3934/cpaa.2020002 [17] 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 [18] Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003 [19] Hiroshi Matsuzawa. A free boundary problem for the Fisher-KPP equation with a given moving boundary. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1821-1852. doi: 10.3934/cpaa.2018087 [20] Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033

2018 Impact Factor: 1.143