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DCDS-S

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.

DCDS-B

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.

DCDS-S

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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