# American Institute of Mathematical Sciences

• Previous Article
Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials
• DCDS Home
• This Issue
• Next Article
Rotating periodic solutions of second order dissipative dynamical systems
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 - A, 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, (1986), 1.   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.  doi: 10.1002/cpa.3160410105.  Google Scholar [3] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, 3rd ed., (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.  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.   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.   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.  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.   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.  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, 23 (2012), 359.   Google Scholar [11] X.-J. Wang, The $k$-Hessian equation,, Geometric analysis and PDEs, (1977), 177.  doi: 10.1007/978-3-642-01674-5_5.  Google Scholar

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, (1986), 1.   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.  doi: 10.1002/cpa.3160410105.  Google Scholar [3] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, 3rd ed., (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.  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.   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.   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.  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.   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.  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, 23 (2012), 359.   Google Scholar [11] X.-J. Wang, The $k$-Hessian equation,, Geometric analysis and PDEs, (1977), 177.  doi: 10.1007/978-3-642-01674-5_5.  Google Scholar
 [1] Tong Tang, Hongjun Gao. Local strong solutions to the compressible viscous magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1617-1633. doi: 10.3934/dcdsb.2016014 [2] Veronica Felli, Ana Primo. Classification of local asymptotics for solutions to heat equations with inverse-square potentials. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 65-107. doi: 10.3934/dcds.2011.31.65 [3] Bingyuan Huang, Shijin Ding, Huanyao Wen. Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1717-1752. doi: 10.3934/dcdss.2016072 [4] Emmanuele DiBenedetto, Ugo Gianazza and Vincenzo Vespri. Intrinsic Harnack estimates for nonnegative local solutions of degenerate parabolic equations. Electronic Research Announcements, 2006, 12: 95-99. [5] Jens Lorenz, Wilberclay G. Melo, Natã Firmino Rocha. The Magneto–Hydrodynamic equations: Local theory and blow-up of solutions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3819-3841. doi: 10.3934/dcdsb.2018332 [6] Tuan Anh Dao, Hironori Michihisa. Study of semi-linear $\sigma$-evolution equations with frictional and visco-elastic damping. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1581-1608. doi: 10.3934/cpaa.2020079 [7] Neil S. Trudinger. On the local theory of prescribed Jacobian equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1663-1681. doi: 10.3934/dcds.2014.34.1663 [8] Kwangseok Choe, Hyungjin Huh. Chern-Simons gauged sigma model into $\mathbb{H}^2$ and its self-dual equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4613-4646. doi: 10.3934/dcds.2019189 [9] Simona Fornaro, Ugo Gianazza. Local properties of non-negative solutions to some doubly non-linear degenerate parabolic equations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 481-492. doi: 10.3934/dcds.2010.26.481 [10] Imran H. Biswas, Indranil Chowdhury. On the differentiability of the solutions of non-local Isaacs equations involving $\frac{1}{2}$-Laplacian. Communications on Pure & Applied Analysis, 2016, 15 (3) : 907-927. doi: 10.3934/cpaa.2016.15.907 [11] Yu-Zhu Wang, Yin-Xia Wang. Local existence of strong solutions to the three dimensional compressible MHD equations with partial viscosity. Communications on Pure & Applied Analysis, 2013, 12 (2) : 851-866. doi: 10.3934/cpaa.2013.12.851 [12] Huxiao Luo, Xianhua Tang, Zu Gao. Sign-changing solutions for non-local elliptic equations with asymptotically linear term. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1147-1159. doi: 10.3934/cpaa.2018055 [13] Xiaojie Hou, Yi Li. Local stability of traveling-wave solutions of nonlinear reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 681-701. doi: 10.3934/dcds.2006.15.681 [14] Thi Tuyen Nguyen. Large time behavior of solutions of local and nonlocal nondegenerate Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator. Communications on Pure & Applied Analysis, 2019, 18 (3) : 999-1021. doi: 10.3934/cpaa.2019049 [15] Hantaek Bae, Rafael Granero-Belinchón, Omar Lazar. On the local and global existence of solutions to 1d transport equations with nonlocal velocity. Networks & Heterogeneous Media, 2019, 14 (3) : 471-487. doi: 10.3934/nhm.2019019 [16] Jae-Myoung  Kim. Local regularity of the magnetohydrodynamics equations near the curved boundary. Communications on Pure & Applied Analysis, 2016, 15 (2) : 507-517. doi: 10.3934/cpaa.2016.15.507 [17] Alberto Ferrero, Filippo Gazzola, Hans-Christoph Grunau. Decay and local eventual positivity for biharmonic parabolic equations. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1129-1157. doi: 10.3934/dcds.2008.21.1129 [18] María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2553-2581. doi: 10.3934/dcdsb.2015.20.2553 [19] Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control & Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61 [20] Wan-Tong Li, Bin-Guo Wang. Attractor minimal sets for nonautonomous type-K competitive and semi-convex delay differential equations with applications. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 589-611. doi: 10.3934/dcds.2009.24.589

2018 Impact Factor: 1.143