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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

 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

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.
Citation: Tiancong Chen, Qing Han. Smooth local solutions to Weingarten equations and $\sigma_k$-equations. Discrete & 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.  Google Scholar [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.  Google Scholar [3] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 3rd ed., Springer, 2001.  Google Scholar [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.  Google Scholar [5] Q. Han, Local solutions to a class of Monge-Ampère equations of the mixed type, Duke Math. J., 136 (2007), 421-473.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [3] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 3rd ed., Springer, 2001.  Google Scholar [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.  Google Scholar [5] Q. Han, Local solutions to a class of Monge-Ampère equations of the mixed type, Duke Math. J., 136 (2007), 421-473.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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