Maximal monotone operators are selfdual vector fields and vice-versa

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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>.