Variational representations for N-cyclically monotone vector fields

Given a convex bounded domain Ω in Rd and an integer N≥2, we associate to any jointly N-monotone (N−1)-tuplet (u1,u2,...,uN−1) of vector fields from into Rd, a Hamiltonian H on Rd×Rd...×Rd, that is concave in the first variable, jointly convex in the last (N−1) variables such that for almost all , \hbox{(u1(x),u2(x),...,uN−1(x))=∇2,...,NH(x,x,...,x). Moreover, H is N-sub-antisymmetric, meaning that ∑i=0N−1H(σi(x))≤0 for all x=(x1,...,xN)∈ΩN, σ being the cyclic permutation on Rd defined by σ(x1,x2,...,xN)=(x2,x3,...,xN,x1). Furthermore, H is N% -antisymmetric in a sense to be defined below. This can be seen as an extension of a theorem of E. Krauss, which associates to any monotone operator, a concave-convex antisymmetric saddle function. We also give various variational characterizations of vector fields that are almost everywhere N-monotone, showing that they are dual to the class of measure preserving N-involutions on Ω.