American Institute of Mathematical Sciences

Advanced
August  2011, 4(4): 887-895. doi: 10.3934/dcdss.2011.4.887

A Liouville-type theorem for some Weingarten hypersurfaces

Shigeru Sakaguchi 1,

1. 

Department of Applied Mathematics, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan

Received  September 2009 Revised  January 2010 Published  November 2010

We consider the entire graph $G$ of a globally Lipschitz continuous function $u$ over $R^N$ with $N \ge 2$, and consider a class of some Weingarten hypersurfaces in $R^{N+1}$. It is shown that, if $u$ solves in the viscosity sense in $R^N$ the fully nonlinear elliptic equation of a Weingarten hypersurface belonging to this class, then $u$ is an affine function and $G$ is a hyperplane. This result is regarded as a Liouville-type theorem for a class of fully nonlinear elliptic equations. The special case for some Monge-Ampère-type equation is related to the previous result of Magnanini and Sakaguchi which gave some characterizations of the hyperplane by making use of stationary isothermic surfaces.
Keywords: Weingarten hypersurfaces, Liouville-type theorem, viscosity solutions..
Mathematics Subject Classification: Primary: 35J60, 53A07; Secondary: 35J1.
Citation: Shigeru Sakaguchi. A Liouville-type theorem for some Weingarten hypersurfaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 887-895. doi: 10.3934/dcdss.2011.4.887
References:
[1]

O. Alvarez, J.-M. Lasry and P.-L. Lions, Convex viscosity solutions and state constraints,, J. Math. Pures Appl., 76 (1997), 265.  doi: doi:10.1016/S0021-7824(97)89952-7.  Google Scholar

[2]

B. Andrews, Pinching estimates and motion of hypersurfaces by curvature functions,, J. Reine Angew. Math., 608 (2007), 17.  doi: doi:10.1515/CRELLE.2007.051.  Google Scholar

[3]

L. Caffarelli, P. Guan and X.-N. Ma, A constant rank theorem for solutions of fully nonlinear elliptic equations,, Comm. Pure Appl. Math., 60 (2007), 1769.  doi: doi:10.1002/cpa.20197.  Google Scholar

[4]

M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations,, Bull. Amer. Math. Soc., 27 (1992), 1.  doi: doi:10.1090/S0273-0979-1992-00266-5.  Google Scholar

[5]

Y. Giga and M. Ohnuma, On strong comparison principle for semicontinuous viscosity solutions of some nonlinear elliptic equations,, Int. J. Pure Appl. Math., 22 (2005), 165.   Google Scholar

[6]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", 2$^{nd}$ edition, (1983).   Google Scholar

[7]

H. Ishii and P.-L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equation,, J. Differential Equations, 83 (1990), 26.  doi: doi:10.1016/0022-0396(90)90068-Z.  Google Scholar

[8]

R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations,, Arch. Rational Mech. Anal., 101 (1988), 1.  doi: doi:10.1007/BF00281780.  Google Scholar

[9]

G. M. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific, (1996).   Google Scholar

[10]

R. Magnanini and S. Sakaguchi, Stationary isothermic surfaces and some characterizations of the hyperplane in the $N$-dimensional Euclidean space,, J. Differential Equations, 248 (2010), 1112.  doi: doi:10.1016/j.jde.2009.11.017.  Google Scholar

[11]

J. Moser, On Harnack's theorem for elliptic differential equations,, Comm. Pure Appl. Math., 14 (1961), 577.  doi: doi:10.1002/cpa.3160140329.  Google Scholar

[12]

N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations,, Arch. Rational Mech. Anal., 111 (1990), 153.  doi: doi:10.1007/BF00375406.  Google Scholar

[13]

J. I. E. Urbas, On the existence of nonclassical solutions for two classes of fully nonlinear elliptic equations,, Indiana Univ. Math. J., 39 (1990), 355.  doi: doi:10.1512/iumj.1990.39.39020.  Google Scholar

show all references

References:
[1]

O. Alvarez, J.-M. Lasry and P.-L. Lions, Convex viscosity solutions and state constraints,, J. Math. Pures Appl., 76 (1997), 265.  doi: doi:10.1016/S0021-7824(97)89952-7.  Google Scholar

[2]

B. Andrews, Pinching estimates and motion of hypersurfaces by curvature functions,, J. Reine Angew. Math., 608 (2007), 17.  doi: doi:10.1515/CRELLE.2007.051.  Google Scholar

[3]

L. Caffarelli, P. Guan and X.-N. Ma, A constant rank theorem for solutions of fully nonlinear elliptic equations,, Comm. Pure Appl. Math., 60 (2007), 1769.  doi: doi:10.1002/cpa.20197.  Google Scholar

[4]

M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations,, Bull. Amer. Math. Soc., 27 (1992), 1.  doi: doi:10.1090/S0273-0979-1992-00266-5.  Google Scholar

[5]

Y. Giga and M. Ohnuma, On strong comparison principle for semicontinuous viscosity solutions of some nonlinear elliptic equations,, Int. J. Pure Appl. Math., 22 (2005), 165.   Google Scholar

[6]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", 2$^{nd}$ edition, (1983).   Google Scholar

[7]

H. Ishii and P.-L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equation,, J. Differential Equations, 83 (1990), 26.  doi: doi:10.1016/0022-0396(90)90068-Z.  Google Scholar

[8]

R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations,, Arch. Rational Mech. Anal., 101 (1988), 1.  doi: doi:10.1007/BF00281780.  Google Scholar

[9]

G. M. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific, (1996).   Google Scholar

[10]

R. Magnanini and S. Sakaguchi, Stationary isothermic surfaces and some characterizations of the hyperplane in the $N$-dimensional Euclidean space,, J. Differential Equations, 248 (2010), 1112.  doi: doi:10.1016/j.jde.2009.11.017.  Google Scholar

[11]

J. Moser, On Harnack's theorem for elliptic differential equations,, Comm. Pure Appl. Math., 14 (1961), 577.  doi: doi:10.1002/cpa.3160140329.  Google Scholar

[12]

N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations,, Arch. Rational Mech. Anal., 111 (1990), 153.  doi: doi:10.1007/BF00375406.  Google Scholar

[13]

J. I. E. Urbas, On the existence of nonclassical solutions for two classes of fully nonlinear elliptic equations,, Indiana Univ. Math. J., 39 (1990), 355.  doi: doi:10.1512/iumj.1990.39.39020.  Google Scholar

[1]

Anh Tuan Duong, Quoc Hung Phan. A Liouville-type theorem for cooperative parabolic systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 823-833. doi: 10.3934/dcds.2018035
[2]

Frank Arthur, Xiaodong Yan, Mingfeng Zhao. A Liouville-type theorem for higher order elliptic systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3317-3339. doi: 10.3934/dcds.2014.34.3317
[3]

Frank Arthur, Xiaodong Yan. A Liouville-type theorem for higher order elliptic systems of Hé non-Lane-Emden type. Communications on Pure & Applied Analysis, 2016, 15 (3) : 807-830. doi: 10.3934/cpaa.2016.15.807
[4]

Quoc Hung Phan. Optimal Liouville-type theorems for a parabolic system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 399-409. doi: 10.3934/dcds.2015.35.399
[5]

Kaouther Ammar, Philippe Souplet. Liouville-type theorems and universal bounds for nonnegative solutions of the porous medium equation with source. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 665-689. doi: 10.3934/dcds.2010.26.665
[6]

Alberto Farina, Miguel Angel Navarro. Some Liouville-type results for stable solutions involving the mean curvature operator: The radial case. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 1233-1256. doi: 10.3934/dcds.2020076
[7]

Pavol Quittner, Philippe Souplet. Parabolic Liouville-type theorems via their elliptic counterparts. Conference Publications, 2011, 2011 (Special) : 1206-1213. doi: 10.3934/proc.2011.2011.1206
[8]

Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855
[9]

Alberto Farina. Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5869-5877. doi: 10.3934/dcds.2015.35.5869
[10]

Philippe Souplet. Liouville-type theorems for elliptic Schrödinger systems associated with copositive matrices. Networks & Heterogeneous Media, 2012, 7 (4) : 967-988. doi: 10.3934/nhm.2012.7.967
[11]

Xiaohui Yu. Liouville type theorem for nonlinear elliptic equation with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4947-4966. doi: 10.3934/dcds.2014.34.4947
[12]

Xinjing Wang, Pengcheng Niu, Xuewei Cui. A Liouville type theorem to an extension problem relating to the Heisenberg group. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2379-2394. doi: 10.3934/cpaa.2018113
[13]

Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511
[14]

Zhenjie Li, Ze Cheng, Dongsheng Li. The Liouville type theorem and local regularity results for nonlinear differential and integral systems. Communications on Pure & Applied Analysis, 2015, 14 (2) : 565-576. doi: 10.3934/cpaa.2015.14.565
[15]

Ovidiu Savin. A Liouville theorem for solutions to the linearized Monge-Ampere equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 865-873. doi: 10.3934/dcds.2010.28.865
[16]

Begoña Barrios, Leandro Del Pezzo, Jorge García-Melián, Alexander Quaas. A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5731-5746. doi: 10.3934/dcds.2017248
[17]

Futoshi Takahashi. Singular extremal solutions to a Liouville-Gelfand type problem with exponential nonlinearity. Conference Publications, 2015, 2015 (special) : 1025-1033. doi: 10.3934/proc.2015.1025
[18]

Zongming Guo, Juncheng Wei. Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2561-2580. doi: 10.3934/dcds.2014.34.2561
[19]

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
[20]

Jingbo Dou, Ye Li. Liouville theorem for an integral system on the upper half space. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 155-171. doi: 10.3934/dcds.2015.35.155

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences