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

A Liouville-type theorem for some Weingarten hypersurfaces

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.
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

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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

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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

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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

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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

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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

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