Variational representations for N-cyclically monotone vector fields

Abstract

Given a convex bounded domain $\Omega $ in ${{\mathbb{R}}}^{d}$ and an integer $N\geq 2$, we associate to any jointly $N$-monotone $(N-1)$-tuplet $(u_1, u_2,..., u_{N-1})$ of vector fields from $% \Omega$ 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 $% x\in \Omega$, \hbox{$(u_1(x), u_2(x),..., u_{N-1}(x))= \nabla_{2,...,N} H(x,x,...,x)$. Moreover, $H$ is $N$-sub-antisymmetric, meaning that $\sum% \limits_{i=0}^{N-1}H(\sigma ^{i}(\mathbf{x}))\leq 0$ for all $\mathbf{x}% =(x_{1},...,x_{N})\in \Omega ^{N}$, $\sigma $ being the cyclic permutation on ${\mathbb{R}}^{d}$ defined by $\sigma (x_{1},x_2,...,x_{N})=(x_{2},x_{3},...,x_{N},x_{1})$. 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 $\Omega$.