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A Liouville-type theorem for some Weingarten hypersurfaces

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


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