Maximal monotone operators are selfdual vector fields and vice-versa

If L is a selfdual Lagrangian L on a reflexive phase space X ×X∗, then the vector field x → ∂̄L(x) := {p ∈ X∗; (p, x) ∈ ∂L(x, p)} is maximal monotone. Conversely, any maximal monotone operator T on X is derived from such a potential on phase space, that is there exists a selfdual Lagrangian L on X ×X∗ (i.e, L∗(p, x) = L(x, p)) such that T = ∂̄L. This solution to problems raised by Fitzpatrick can be seen as an extension of a celebrated result of Rockafellar stating that maximal cyclically monotone operators are actually of the form T = ∂φ for some convex lower semi-continuous function on X. This representation allows for the application of the selfdual variational theory –recently developed by the author– to the equations driven by maximal monotone vector fields. Consequently, solutions to equations of the form Λx ∈ Tx for a given map Λ : D(Λ) ⊂ X → X∗, can now be obtained by minimizing functionals of the form I(x) = L(x,Λx)−〈x,Λx〉.