Variational representations for N-cyclically monotone vector fields

Given a convex bounded domain $\mathrm{\Omega }$ in ${\mathbb{R}}^d$ and an integer $N\ge 2$, we associate to any jointly $N$-monotone $(N-1)$-tuplet $(u_1,u_2,\mathrm{.}\mathrm{.}\mathrm{.},u_{N-1})$ of vector fields from $$ into ${\mathbb{R}}^d$, a Hamiltonian $H$ on ${$\mathbb{R}$}^{d} \times {$\mathbb{R}$}^{d} … \times {$\mathbb{R}$}^{d}$, that is concave in the first variable, jointly convex in the last $(N-1)$ variables such that for almost all $$, \hbox{$(u_1(x), u_2(x),…, u_{N-1}(x))= \nabla_{2,…,N} H(x,x,…,x)$\mathrm{.}Moreover,$H$is$N$-sub-antisymmetric,meaningthat$\sum% \limits_{i=0}^{N-1}H(\sigma ^{i}($\mathbf{x}$))\leq 0$forall$$\mathbf{x}$% =(x_{1},…,x_{N})\in \Omega ^{N}$,$\sigma $beingthecyclicpermutationon${$\mathbb{R}$}^{d}$definedby$\sigma (x_{1},x_2,…,x_{N})=(x_{2},x_{3},…,x_{N},x_{1})$\mathrm{.}Furthermore,$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 $\mathrm{\Omega }$.