# Variational representations for N-cyclically monotone vector fields

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Variational representations for N-cyclically monotone vector fields**

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 1x, u 2x,

., u {N-1}x= abla {2,

.,N} Hx,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 $\Omega$.

Author: **Alfred Galichon; Nassif Ghoussoub**

Source: https://archive.org/