A Liouville-type theorem for some Weingarten hypersurfaces

Shigeru Sakaguchi

doi:10.3934/dcdss.2011.4.887

Article Contents

2011, Volume 4, Issue 4: 887-895. Doi: 10.3934/dcdss.2011.4.887

This issue Previous Article The degenerate drift-diffusion system with the Sobolev critical exponent Next Article Singular backward self-similar solutions of a semilinear parabolic equation

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

Received: August 31, 2009

Revised: December 31, 2009

Published: July 2011

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Abstract

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.

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

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