Variational representations for N-cyclically monotone vector fields

Abstract

Given a convex bounded domain $\Omega$ in ${\mathbb{R}}^d$ and an integer $N\ge 2$, we associate to any jointly $N$-monotone $(N-1)$-tuplet $(u_1,u_2,...,u_{N-1})$ 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{$(u_1(x),u_2(x),...,u_{N-1}(x))={\nabla }_{2,...,N}H(x,x,...,x)$. Moreover, $H$ is $N$-sub-antisymmetric, meaning that $\sum$ for all $\mathbf{x}$, $\sigma$ 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 $\Omega$.

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