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$C^\infty$ 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 |
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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. |
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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. |
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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1983.
doi: 10.1007/978-3-642-61798-0. |
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Q. Han, Local solutions to a class of Monge-Ampère equations of mixed type, Duke Math. J., 136 (2007), 421-474. |
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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. |
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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. |
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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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