# American Institute of Mathematical Sciences

• Previous Article
Free boundary problems associated with cancer treatment by combination therapy
• DCDS Home
• This Issue
• Next Article
The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals
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 & Continuous Dynamical Systems - A, 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. Google Scholar [2] L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, Colloquium Publications, 43, American Mathematical Society, Providence, RI, 1995. doi: 10.1090/coll/043. Google Scholar [3] L. Caffarelli, L. 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. Google Scholar [4] L. Caffarelli, L. 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. Google Scholar [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. Google Scholar [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. Google Scholar [7] P. L. Lions, Sur les équations de Monge-Ampère. Ⅰ, Manuscripta Math., 41 (1983), 1-43. doi: 10.1007/BF01165928. Google Scholar [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. Google Scholar

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. Google Scholar [2] L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, Colloquium Publications, 43, American Mathematical Society, Providence, RI, 1995. doi: 10.1090/coll/043. Google Scholar [3] L. Caffarelli, L. 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. Google Scholar [4] L. Caffarelli, L. 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. Google Scholar [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. Google Scholar [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. Google Scholar [7] P. L. Lions, Sur les équations de Monge-Ampère. Ⅰ, Manuscripta Math., 41 (1983), 1-43. doi: 10.1007/BF01165928. Google Scholar [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. Google Scholar
 [1] 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 [2] Imed Bachar, Habib Mâagli. Singular solutions of a nonlinear equation in a punctured domain of $\mathbb{R}^{2}$. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 171-188. doi: 10.3934/dcdss.2019012 [3] Yongkuan Cheng, Yaotian Shen. Generalized quasilinear Schrödinger equations with concave functions $l(s^2)$. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1311-1343. doi: 10.3934/dcds.2019056 [4] Peng Mei, Zhan Zhou, Genghong Lin. Periodic and subharmonic solutions for a 2$n$th-order $\phi_c$-Laplacian difference equation containing both advances and retardations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2085-2095. doi: 10.3934/dcdss.2019134 [5] Xiaopeng Zhao. Space-time decay estimates of solutions to liquid crystal system in $\mathbb{R}^3$. Communications on Pure & Applied Analysis, 2019, 18 (1) : 1-13. doi: 10.3934/cpaa.2019001 [6] Yu-Zhao Wang. $\mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116 [7] K. D. Chu, D. D. Hai. Positive solutions for the one-dimensional singular superlinear $p$-Laplacian problem. Communications on Pure & Applied Analysis, 2020, 19 (1) : 241-252. doi: 10.3934/cpaa.2020013 [8] Phuong Le. Symmetry of singular solutions for a weighted Choquard equation involving the fractional $p$-Laplacian. Communications on Pure & Applied Analysis, 2020, 19 (1) : 527-539. doi: 10.3934/cpaa.2020026 [9] Linlin Fu, Jiahao Xu. A new proof of continuity of Lyapunov exponents for a class of $C^2$ quasiperiodic Schrödinger cocycles without LDT. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2915-2931. doi: 10.3934/dcds.2019121 [10] Shengbing Deng. Construction solutions for Neumann problem with Hénon term in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2233-2253. doi: 10.3934/dcds.2019094 [11] Guoyuan Chen, Yong Liu, Juncheng Wei. Nondegeneracy of harmonic maps from ${{\mathbb{R}}^{2}}$ to ${{\mathbb{S}}^{2}}$. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-19. doi: 10.3934/dcds.2019228 [12] Genghong Lin, Zhan Zhou. Homoclinic solutions of discrete $\phi$-Laplacian equations with mixed nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1723-1747. doi: 10.3934/cpaa.2018082 [13] Abdelwahab Bensouilah, Van Duong Dinh, Mohamed Majdoub. Scattering in the weighted $L^2$-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2735-2755. doi: 10.3934/cpaa.2019122 [14] Kwangseok Choe, Hyungjin Huh. Chern-Simons gauged sigma model into $\mathbb{H}^2$ and its self-dual equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4613-4646. doi: 10.3934/dcds.2019189 [15] Qianying Xiao, Zuohuan Zheng. $C^1$ weak Palis conjecture for nonsingular flows. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1809-1832. doi: 10.3934/dcds.2018074 [16] Ilwoo Cho, Palle Jorgense. Free probability on $C^{*}$-algebras induced by hecke algebras over primes. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 2221-2252. doi: 10.3934/dcdss.2019143 [17] Florin Diacu, Shuqiang Zhu. Almost all 3-body relative equilibria on $\mathbb S^2$ and $\mathbb H^2$ are inclined. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-13. doi: 10.3934/dcdss.2020067 [18] Joackim Bernier. Bounds on the growth of high discrete Sobolev norms for the cubic discrete nonlinear Schrödinger equations on $h\mathbb{Z}$. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3179-3195. doi: 10.3934/dcds.2019131 [19] Tingting Wu, Jian Gao, Yun Gao, Fang-Wei Fu. ${{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 641-657. doi: 10.3934/amc.2018038 [20] Tuan Anh Dao, Michael Reissig. $L^1$ estimates for oscillating integrals and their applications to semi-linear models with $\sigma$-evolution like structural damping. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5431-5463. doi: 10.3934/dcds.2019222

2018 Impact Factor: 1.143

Article outline