  Let us consider the autonomous obstacle problem \begin{equation*} \min_v
\int_\Omega F(Dv(x)) \, dx \end{equation*} on a specific class of admissible
functions, where we suppose the Lagrangian satisfies proper hypotheses of
convexity and superlinearity at infinity. Our aim is to characterize the
solution, which exists and it is unique, thanks to a primal-dual formulation of
the problem. The proof is based on classical arguments of Convex Analysis and
on Calculus of Variations' techniques.
