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

Previous Article
Boundary behavior and asymptotic behavior of solutions to a class of parabolic equations with boundary degeneracy
DCDS Home
This Issue
Next Article
Existence and stability of a two-parameter family of solitary waves for a 2-coupled nonlinear Schrödinger system

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

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

Guji Tian ^1, , Qi Wang ^2, and Chao-Jiang Xu ^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

Abstract
Related Papers

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.

Keywords: convexities., Elliptical $k$- Hessian equation, existence of local solutions.

Mathematics Subject Classification: Primary: 35J60; Secondary: 35J7.

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.

[1]	Kods Hassine. Existence and uniqueness of radial solutions for Hardy-Hénon equations involving k-Hessian operators. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022084
[2]	Md. Rabiul Haque, Takayoshi Ogawa, Ryuichi Sato. Existence of weak solutions to a convection–diffusion equation in a uniformly local lebesgue space. Communications on Pure and Applied Analysis, 2020, 19 (2) : 677-697. doi: 10.3934/cpaa.2020031
[3]	Giovany M. Figueiredo, Tarcyana S. Figueiredo-Sousa, Cristian Morales-Rodrigo, Antonio Suárez. Existence of positive solutions of an elliptic equation with local and nonlocal variable diffusion coefficient. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3689-3711. doi: 10.3934/dcdsb.2018311
[4]	Xie Li, Zhaoyin Xiang. Existence and nonexistence of local/global solutions for a nonhomogeneous heat equation. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1465-1480. doi: 10.3934/cpaa.2014.13.1465
[5]	Tiancong Chen, Qing Han. Smooth local solutions to Weingarten equations and $\sigma_k$-equations. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 653-660. doi: 10.3934/dcds.2016.36.653
[6]	Borys Alvarez-Samaniego, Pascal Azerad. Existence of travelling-wave solutions and local well-posedness of the Fowler equation. Discrete and Continuous Dynamical Systems - B, 2009, 12 (4) : 671-692. doi: 10.3934/dcdsb.2009.12.671
[7]	Nicholas J. Kass, Mohammad A. Rammaha. Local and global existence of solutions to a strongly damped wave equation of the $ p $-Laplacian type. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1449-1478. doi: 10.3934/cpaa.2018070
[8]	Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Local existence with mild regularity for the Boltzmann equation. Kinetic and Related Models, 2013, 6 (4) : 1011-1041. doi: 10.3934/krm.2013.6.1011
[9]	Limei Dai. Existence and nonexistence of subsolutions for augmented Hessian equations. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 579-596. doi: 10.3934/dcds.2020023
[10]	Alexander V. Kolesnikov. Hessian metrics, $CD(K,N)$-spaces, and optimal transportation of log-concave measures. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1511-1532. doi: 10.3934/dcds.2014.34.1511
[11]	Rui Liu. Some new results on explicit traveling wave solutions of $K(m, n)$ equation. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 633-646. doi: 10.3934/dcdsb.2010.13.633
[12]	Sylvie Oliffson Kamphorst, Sônia Pinto de Carvalho. Elliptic islands on the elliptical stadium. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 663-674. doi: 10.3934/dcds.2001.7.663
[13]	Xinqun Mei, Jundong Zhou. The interior gradient estimate of prescribed Hessian quotient curvature equation in the hyperbolic space. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1187-1198. doi: 10.3934/cpaa.2021012
[14]	Feida Jiang, Xi Chen, Juhua Shi. Nonexistence of entire positive solutions for conformal Hessian quotient inequalities. Electronic Research Archive, 2021, 29 (6) : 4075-4086. doi: 10.3934/era.2021072
[15]	H. T. Liu. Impulsive effects on the existence of solutions for a fast diffusion equation. Conference Publications, 2001, 2001 (Special) : 248-253. doi: 10.3934/proc.2001.2001.248
[16]	Changchun Liu, Jingxue Yin, Juan Zhou. Existence of weak solutions for a generalized thin film equation. Communications on Pure and Applied Analysis, 2007, 6 (2) : 465-480. doi: 10.3934/cpaa.2007.6.465
[17]	Andrea L. Bertozzi, Dejan Slepcev. Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1617-1637. doi: 10.3934/cpaa.2010.9.1617
[18]	Kin Ming Hui, Sunghoon Kim. Existence of Neumann and singular solutions of the fast diffusion equation. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4859-4887. doi: 10.3934/dcds.2015.35.4859
[19]	Antonio Greco, Antonio Iannizzotto. Existence and convexity of solutions of the fractional heat equation. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2201-2226. doi: 10.3934/cpaa.2017109
[20]	Xiaoming Fu, Quentin Griette, Pierre Magal. Existence and uniqueness of solutions for a hyperbolic Keller–Segel equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 1931-1966. doi: 10.3934/dcdsb.2020326

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Guji Tian Qi Wang Chao-Jiang Xu
on Google Scholar
Guji Tian Qi Wang Chao-Jiang Xu

[Back to Top]