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 plug 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.)