On the maximal monotonicity of diagonal subdifferential operators

Consider a real-valued bifunction f which is concave in its first argument and convex in its second one. We study its subdifferential with respect to the second argument, evaluated at pairs of the form (x, x), and the subdifferential of −f with respect to its first argument, evaluated at the same pairs. The resulting operators are not always monotone, and we analyze additional conditions on f which ensure their monotonicity, and furthermore their maximal monoticity. Our main result is that these operators are maximal monotone when f is continuous and it vanishes whenever both arguments coincide. Our results have consequences in terms of the reformulation of equilibrium problems as variational inequality ones.