Smooth local solutions to Weingarten equations and \sigma_k-equations

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February 2016, 36(2): 653-660. doi: 10.3934/dcds.2016.36.653

Smooth local solutions to Weingarten equations and $\sigma_k$-equations

Tiancong Chen ^1, and Qing Han ^2,

1.	Department of Mathematics, University of California, Santa Barbara, CA 93106, United States
2.	Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States

Received June 2014 Revised March 2015 Published August 2015

Abstract
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In this paper, we study the existence of smooth local solutions to Weingarten equations and $\sigma_k$-equations. We will prove that, for $2\le k\le n-1$, the Weingarten equations and the $\sigma_k$-equations always have smooth local solutions regardless of the sign of the functions in the right-hand side of the equations. We will demonstrate that the associated linearized equations are uniformly elliptic if we choose the initial approximate solutions appropriately.

Keywords: $\sigma_k$-equations, Weingarten equations, local solutions..

Mathematics Subject Classification: Primary: 35A01, 35J6.

Citation: 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

References:

[1]	L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second-order elliptic equations, IV. Starshaped compact Weingarten hypersurfaces, Current topics in partial differential equations, Kinokuniya, Tokyo, (1986), 1-26.
[2]	L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second-order elliptic equations V. The Dirichlet problem for Weingarten Hypersurfaces, Comm. Pure Appl. Math., 41 (1988), 47-70. doi: 10.1002/cpa.3160410105.
[3]	D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 3rd ed., Springer, 2001.
[4]	Q. Han, On the isometric embedding of surfaces with Gauss curvature changing sign cleanly, Comm. Pure Appl. Math., 58 (2005), 285-295. doi: 10.1002/cpa.20054.
[5]	Q. Han, Local solutions to a class of Monge-Ampère equations of the mixed type, Duke Math. J., 136 (2007), 421-473.
[6]	Q. Han, J.-X. Hong and C.-S. Lin, Local isometric embedding of surfaces with nonpositive gaussian curvature, J. Diff. Geometry, 63 (2003), 475-520.
[7]	J.-X. Hong and C. Zuily, Existence of $C^\infty$ local solutions for the Monge-Ampère equation, Invent. Math., 89 (1987), 645-661. doi: 10.1007/BF01388988.
[8]	C.-S. Lin, The local isometric embedding in $\mathbbR^3$ of 2-dimensional Riemannian manifolds with nonnegative curvature, J. Diff. Geometry, 21 (1985), 213-230.
[9]	C.-S. Lin, The local isometric embedding in $\mathbbR^3$ of two dimensional Riemannian manifolds with Gaussian curvature changing sign clearly, Comm. Pure Appl. Math., 39 (1986), 867-887. doi: 10.1002/cpa.3160390607.
[10]	W.-M. Sheng, N. Trudinger and X.-J. Wang, Prescribed Weingarten curvature equations, Recent Development in Geometry and Analysis, International Press, 23 (2012), 359-386.
[11]	X.-J. Wang, The $k$-Hessian equation, Geometric analysis and PDEs, 177-252, Lecture Notes in Math., 1977, Springer, 2009. doi: 10.1007/978-3-642-01674-5_5.

show all references

References:

[1]	L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second-order elliptic equations, IV. Starshaped compact Weingarten hypersurfaces, Current topics in partial differential equations, Kinokuniya, Tokyo, (1986), 1-26.
[2]	L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second-order elliptic equations V. The Dirichlet problem for Weingarten Hypersurfaces, Comm. Pure Appl. Math., 41 (1988), 47-70. doi: 10.1002/cpa.3160410105.
[3]	D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 3rd ed., Springer, 2001.
[4]	Q. Han, On the isometric embedding of surfaces with Gauss curvature changing sign cleanly, Comm. Pure Appl. Math., 58 (2005), 285-295. doi: 10.1002/cpa.20054.
[5]	Q. Han, Local solutions to a class of Monge-Ampère equations of the mixed type, Duke Math. J., 136 (2007), 421-473.
[6]	Q. Han, J.-X. Hong and C.-S. Lin, Local isometric embedding of surfaces with nonpositive gaussian curvature, J. Diff. Geometry, 63 (2003), 475-520.
[7]	J.-X. Hong and C. Zuily, Existence of $C^\infty$ local solutions for the Monge-Ampère equation, Invent. Math., 89 (1987), 645-661. doi: 10.1007/BF01388988.
[8]	C.-S. Lin, The local isometric embedding in $\mathbbR^3$ of 2-dimensional Riemannian manifolds with nonnegative curvature, J. Diff. Geometry, 21 (1985), 213-230.
[9]	C.-S. Lin, The local isometric embedding in $\mathbbR^3$ of two dimensional Riemannian manifolds with Gaussian curvature changing sign clearly, Comm. Pure Appl. Math., 39 (1986), 867-887. doi: 10.1002/cpa.3160390607.
[10]	W.-M. Sheng, N. Trudinger and X.-J. Wang, Prescribed Weingarten curvature equations, Recent Development in Geometry and Analysis, International Press, 23 (2012), 359-386.
[11]	X.-J. Wang, The $k$-Hessian equation, Geometric analysis and PDEs, 177-252, Lecture Notes in Math., 1977, Springer, 2009. doi: 10.1007/978-3-642-01674-5_5.

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