  We consider stable solutions of a semilinear elliptic equation with
homogeneous Neumann boundary conditions. A classical result of Casten, Holland
[20] and Matano [44] states that all stable solutions are constant in convex
bounded domains. In this paper, we examine whether this result extends to
unbounded convex domains. We give a positive answer for stable non-degenerate
solutions, and for stable solutions if the domain $\Omega$ further satisfies
$\Omega$ $\cap$ {|x| $\le$ R} = O(R^2), when R $\rightarrow$ +$\infty$. If the
domain is a straight cylinder, an additional natural assumption is needed.
These results can be seen as an extension to more general domains of some
results on De Giorgi's conjecture.As an application, we establish asymptotic
symmetries for stable solutions when the domain satisfies a geometric property
asymptotically.
