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

Connor Mooney; Ovidiu Savin

doi:10.3934/dcds.2019235

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2019, Volume 39, Issue 12: 6865-6876. Doi: 10.3934/dcds.2019235

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

Connor Mooney^1, and
Ovidiu Savin^2, ,

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
^* Corresponding author: Ovidiu Savin

Received: July 2018

Revised: October 2018

Published online: December 2019

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.

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Abstract

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.

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Mathematics Subject Classification: Primary: 35J96, 35B65.

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References

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