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Article Contents

# A Liouville-type theorem for some Weingarten hypersurfaces

• 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.
Mathematics Subject Classification: Primary: 35J60, 53A07; Secondary: 35J15.

 Citation:

•  [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.