A Liouville-type theorem for some Weingarten hypersurfaces

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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

Abstract
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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 and 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), 265-288. doi: doi:10.1016/S0021-7824(97)89952-7.
[2]	B. Andrews, Pinching estimates and motion of hypersurfaces by curvature functions, J. Reine Angew. Math., 608 (2007), 17-33. doi: doi:10.1515/CRELLE.2007.051.
[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), 1769-1791. doi: doi:10.1002/cpa.20197.
[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), 1-67. doi: doi:10.1090/S0273-0979-1992-00266-5.
[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), 165-184.
[6]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd edition, Springer-Verlag, Berlin, Heidelberg, New York, 1983.
[7]	H. Ishii and P.-L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equation, J. Differential Equations, 83 (1990), 26-78. doi: doi:10.1016/0022-0396(90)90068-Z.
[8]	R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rational Mech. Anal., 101 (1988), 1-27. doi: doi:10.1007/BF00281780.
[9]	G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific, 1996.
[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), 1112-1119. doi: doi:10.1016/j.jde.2009.11.017.
[11]	J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591. doi: doi:10.1002/cpa.3160140329.
[12]	N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Rational Mech. Anal., 111 (1990), 153-179. doi: doi:10.1007/BF00375406.
[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), 355-382. doi: doi:10.1512/iumj.1990.39.39020.

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), 265-288. doi: doi:10.1016/S0021-7824(97)89952-7.
[2]	B. Andrews, Pinching estimates and motion of hypersurfaces by curvature functions, J. Reine Angew. Math., 608 (2007), 17-33. doi: doi:10.1515/CRELLE.2007.051.
[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), 1769-1791. doi: doi:10.1002/cpa.20197.
[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), 1-67. doi: doi:10.1090/S0273-0979-1992-00266-5.
[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), 165-184.
[6]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd edition, Springer-Verlag, Berlin, Heidelberg, New York, 1983.
[7]	H. Ishii and P.-L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equation, J. Differential Equations, 83 (1990), 26-78. doi: doi:10.1016/0022-0396(90)90068-Z.
[8]	R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rational Mech. Anal., 101 (1988), 1-27. doi: doi:10.1007/BF00281780.
[9]	G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific, 1996.
[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), 1112-1119. doi: doi:10.1016/j.jde.2009.11.017.
[11]	J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591. doi: doi:10.1002/cpa.3160140329.
[12]	N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Rational Mech. Anal., 111 (1990), 153-179. doi: doi:10.1007/BF00375406.
[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), 355-382. doi: doi:10.1512/iumj.1990.39.39020.

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