December  2019, 39(12): 6865-6876. doi: 10.3934/dcds.2019235

Regularity results for the equation $ u_{11}u_{22} = 1 $

1. 

410C Rowland Hall, UC Irvine, Irvine, CA 92697-3875, USA

2. 

Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027, USA

* Corresponding author: Ovidiu Savin

Received  July 2018 Revised  October 2018 Published  June 2019

Fund Project: C. Mooney was supported by NSF grant DMS-1501152 and ERC grant "Regularity and Stability in Partial Differential Equations" (RSPDE). O. Savin was supported by NSF grant DMS-1500438.

We study the equation $u_{11}u_{22} = 1$ in $\mathbb{R}^2$. Our results include an interior $C^2$ estimate, classical solvability of the Dirichlet problem, and the existence of non-quadratic entire solutions. We also construct global singular solutions to the analogous equation in higher dimensions. At the end we state some open questions.

Citation: Connor Mooney, Ovidiu Savin. Regularity results for the equation $ u_{11}u_{22} = 1 $. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 6865-6876. doi: 10.3934/dcds.2019235
References:
[1]

Z. Błocki, On the regularity of the complex Monge-Ampère operator, Complex Geometric Analysis in Pohang, 222 (1997), 181-189.  doi: 10.1090/conm/222/03161.

[2]

L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, Colloquium Publications, 43, American Mathematical Society, Providence, RI, 1995. doi: 10.1090/coll/043.

[3]

L. CaffarelliL. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. Ⅰ. Monge-Ampère equation, Comm. Pure Appl. Math., 37 (1984), 369-402.  doi: 10.1002/cpa.3160370306.

[4]

L. CaffarelliL. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. Ⅲ. Functions of the eigenvalues of the Hessian, Acta Math., 155 (1985), 261-301.  doi: 10.1007/BF02392544.

[5]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1983. doi: 10.1007/978-3-642-61798-0.

[6]

H. Ishii, On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDE's, Comm. Pure Appl. Math., 42 (1989), 15-45.  doi: 10.1002/cpa.3160420103.

[7]

P. L. Lions, Sur les équations de Monge-Ampère. Ⅰ, Manuscripta Math., 41 (1983), 1-43.  doi: 10.1007/BF01165928.

[8]

A. Pogorelov, The regularity of the generalized solutions of the equation $\det\left(\frac{\partial^2 u}{\partial x_i\partial x_j}\right) = \varphi(x_1, x_2, ..., x_n) > 0$, Dokl. Akad. Nauk SSSR, 200 (1971), 534-537. 

show all references

References:
[1]

Z. Błocki, On the regularity of the complex Monge-Ampère operator, Complex Geometric Analysis in Pohang, 222 (1997), 181-189.  doi: 10.1090/conm/222/03161.

[2]

L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, Colloquium Publications, 43, American Mathematical Society, Providence, RI, 1995. doi: 10.1090/coll/043.

[3]

L. CaffarelliL. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. Ⅰ. Monge-Ampère equation, Comm. Pure Appl. Math., 37 (1984), 369-402.  doi: 10.1002/cpa.3160370306.

[4]

L. CaffarelliL. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. Ⅲ. Functions of the eigenvalues of the Hessian, Acta Math., 155 (1985), 261-301.  doi: 10.1007/BF02392544.

[5]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1983. doi: 10.1007/978-3-642-61798-0.

[6]

H. Ishii, On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDE's, Comm. Pure Appl. Math., 42 (1989), 15-45.  doi: 10.1002/cpa.3160420103.

[7]

P. L. Lions, Sur les équations de Monge-Ampère. Ⅰ, Manuscripta Math., 41 (1983), 1-43.  doi: 10.1007/BF01165928.

[8]

A. Pogorelov, The regularity of the generalized solutions of the equation $\det\left(\frac{\partial^2 u}{\partial x_i\partial x_j}\right) = \varphi(x_1, x_2, ..., x_n) > 0$, Dokl. Akad. Nauk SSSR, 200 (1971), 534-537. 

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