A Liouville-type theorem for some Weingarten hypersurfaces
Shigeru Sakaguchi
Discrete & Continuous Dynamical Systems - S 2011, 4(4): 887-895 doi: 10.3934/dcdss.2011.4.887
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.
A linear-quadratic control problem with discretionary stopping
Shigeaki Koike Hiroaki Morimoto Shigeru Sakaguchi
Discrete & Continuous Dynamical Systems - B 2007, 8(2): 261-277 doi: 10.3934/dcdsb.2007.8.261
We study the variational inequality for a 1-dimensional linear-quadratic control problem with discretionary stopping. We establish the existence of a unique strong solution via stochastic analysis and the viscosity solution technique. Finally, the optimal policy is shown to exist from the optimality conditions.
keywords: Viscosity solution stopping time linear-quadratic control.
Filippo Gazzola Rolando Magnanini Shigeru Sakaguchi
Discrete & Continuous Dynamical Systems - S 2011, 4(4): i-ii doi: 10.3934/dcdss.2011.4.4i
Qualitative aspects of parabolic and elliptic partial differential equations have attracted much attention from the early beginnings. In recent years, once basic issues about PDE's, such as existence, uniqueness, stability and regularity of solutions of initial/boundary value problems, have been quite understood, research on topological and/or geometric properties of their solutions have become more intense.

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