To start, a set $ G\in \mathbb R^2$ is called maximal monotone if it is not a strict subset of a monotone set. For each maximal monotone set $ G$ , we can define a function $ $ u_G(x_1,x_2)=\inf_{y\in G} \{(x_1-y_1)(x_2-y_2)\}-x_1x_2$ $ The collection of all such functions will be denoted by $ \mathcal U$ . It is easy to see that every $ u\in\mathcal U$ is concave, and $ u(x)\leq -x_1x_2$ . Heuristically, if we apply envelope theorem to differentiate $ u$ ,we should expect $ $ \partial_{x_1}u(x)=-y_2\ \partial_{x_2}u(x)=-y_1 $ $ where $ (y_1,y_2)$ is a point such that the infimum is attained. It follows by plugging such $ y$ into the infimum that $ $ u(x)= \partial_{x_1}u(x)\partial_{x_2}u(x)+x_1\partial_{x_1}u(x)+x_2\partial_{x_2}u(x)$ $

My question: Is there a characterization of the family $ \mathcal U$ , which is independent of the notion of maximal monotone set? (For example, I want to say $ \mathcal U$ contains all concave functions bounded from above by $ -x_1x_2$ , such that some differential(sub-differential) equations/inclusions is satisfied.)