Variational representations for N-cyclically monotone vector fields

TL;DR

This paper extends the theory of monotone vector fields by associating them with Hamiltonians that are concave in one variable and convex in others, revealing new dualities and variational characterizations.

Contribution

It introduces a variational framework linking N-cyclically monotone vector fields to Hamiltonians with specific symmetry and convexity properties, generalizing Krauss's theorem.

Findings

Constructs Hamiltonians for N-cyclically monotone vector fields.

Establishes duality between almost everywhere N-monotone fields and measure-preserving N-involutions.

Provides variational characterizations of N-monotonicity.

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 $\Omega$.