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February  2016, 36(2): 1023-1039. doi: 10.3934/dcds.2016.36.1023

$C^\infty$ Local solutions of elliptical $2-$Hessian equation in $\mathbb{R}^3$

1. 

Wuhan Institute of Physics and Mathematics,Chinese Academy of Sciences, Wuhan 430071

2. 

School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

3. 

School of Mathematics, Wuhan University, 430072, Wuhan

Received  June 2014 Revised  March 2015 Published  August 2015

In this work, we study the existence of $C^{\infty}$ local solutions to $2$-Hessian equation in $\mathbb{R}^{3}$. We consider the case that the right hand side function $f$ possibly vanishes, changes the sign, is positively or negatively defined. We also give the convexities of solutions which are related with the annulation or the sign of right-hand side function $f$. The associated linearized operator are uniformly elliptic.
Citation: Guji Tian, Qi Wang, Chao-Jiang Xu. $C^\infty$ Local solutions of elliptical $2-$Hessian equation in $\mathbb{R}^3$. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1023-1039. doi: 10.3934/dcds.2016.36.1023
References:
[1]

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

[2]

B. Guan and J. Spruck, Locally convex hypersurfaces of constant curvature with boundary, Comm. Pure Appl. Math., 57 (2004), 1311-1331. doi: 10.1002/cpa.20010.

[3]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1983. doi: 10.1007/978-3-642-61798-0.

[4]

Q. Han, Local solutions to a class of Monge-Ampère equations of mixed type, Duke Math. J., 136 (2007), 421-474.

[5]

J. Hong and C. Zuily, Exitence of $C^{\infty}$ local solutions for the Monge-Ampère equation, Invent.Math., 89 (1987), 645-661. doi: 10.1007/BF01388988.

[6]

N. M. Ivochkina, A description of the stability cones generated by differential operators of Monge-Ampère type, English transl., Math. USSR Sbornik, 50 (1985), 259-268.

[7]

N. M. Ivochkina, S. I. Prokofeva and G. V. Yakunina, The Gårding cones in the modern theory of fully nonlinear second order differential equations, Journal of Mathematical Sciences, 184 (2012), 295-315. doi: 10.1007/s10958-012-0869-1.

[8]

C. S. Lin, The local isometric embedding in $\mathbbR^3$ of 2-dimensional Riemannian manifolds with non negative curvature, Journal of Diff. Equations, 21 (1985), 213-230.

[9]

C. S. Lin, The local isometric embedding in $\mathbbR^{3}$ of two dimensinal Riemannian manifolds with Gaussian curvature changing sign clearly, Comm. Pure Appl. Math., 39 (1986), 867-887. doi: 10.1002/cpa.3160390607.

[10]

Q. Wang and C.-J. Xu, $C^{1,1}$ solution of the Dirichlet problem for degenerate k-Hessian equations, Nonlinear Analysis, T. M. A., 104 (2014), 133-146. doi: 10.1016/j.na.2014.03.016.

[11]

X.-N. Ma and L. Xu, The convexity of solution of a class Hessian equation in bounded convex domain in $\mathbbR^3$, Journal of Funct. Anal., 255 (2008), 1713-1723. doi: 10.1016/j.jfa.2008.06.008.

[12]

X.-J. Wang, The $k$-Hessian equation, Geometric Analysis and PDEs, Lecture Notes in Mathematics, 1977 (2009), 177-252. doi: 10.1007/978-3-642-01674-5_5.

show all references

References:
[1]

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

[2]

B. Guan and J. Spruck, Locally convex hypersurfaces of constant curvature with boundary, Comm. Pure Appl. Math., 57 (2004), 1311-1331. doi: 10.1002/cpa.20010.

[3]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1983. doi: 10.1007/978-3-642-61798-0.

[4]

Q. Han, Local solutions to a class of Monge-Ampère equations of mixed type, Duke Math. J., 136 (2007), 421-474.

[5]

J. Hong and C. Zuily, Exitence of $C^{\infty}$ local solutions for the Monge-Ampère equation, Invent.Math., 89 (1987), 645-661. doi: 10.1007/BF01388988.

[6]

N. M. Ivochkina, A description of the stability cones generated by differential operators of Monge-Ampère type, English transl., Math. USSR Sbornik, 50 (1985), 259-268.

[7]

N. M. Ivochkina, S. I. Prokofeva and G. V. Yakunina, The Gårding cones in the modern theory of fully nonlinear second order differential equations, Journal of Mathematical Sciences, 184 (2012), 295-315. doi: 10.1007/s10958-012-0869-1.

[8]

C. S. Lin, The local isometric embedding in $\mathbbR^3$ of 2-dimensional Riemannian manifolds with non negative curvature, Journal of Diff. Equations, 21 (1985), 213-230.

[9]

C. S. Lin, The local isometric embedding in $\mathbbR^{3}$ of two dimensinal Riemannian manifolds with Gaussian curvature changing sign clearly, Comm. Pure Appl. Math., 39 (1986), 867-887. doi: 10.1002/cpa.3160390607.

[10]

Q. Wang and C.-J. Xu, $C^{1,1}$ solution of the Dirichlet problem for degenerate k-Hessian equations, Nonlinear Analysis, T. M. A., 104 (2014), 133-146. doi: 10.1016/j.na.2014.03.016.

[11]

X.-N. Ma and L. Xu, The convexity of solution of a class Hessian equation in bounded convex domain in $\mathbbR^3$, Journal of Funct. Anal., 255 (2008), 1713-1723. doi: 10.1016/j.jfa.2008.06.008.

[12]

X.-J. Wang, The $k$-Hessian equation, Geometric Analysis and PDEs, Lecture Notes in Mathematics, 1977 (2009), 177-252. doi: 10.1007/978-3-642-01674-5_5.

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