Variational representations for $N$-cyclically monotone vector fields

Alfred Galichon and Nassif Ghoussoub

Vol. 269 (2014), No. 2, 323–340

DOI: 10.2140/pjm.2014.269.323

Abstract

Given a convex bounded domain $\Omega$ in ${ℝ}^d$ and an integer $N\ge 2$, we associate to any jointly $N$-monotone $\left(N-1\right)$-tuplet $\left(u_1,u_2,\dots ,u_{N-1}\right)$ of vector fields from $\Omega$ into ${ℝ}^d$ a Hamiltonian $H$ on ${ℝ}^d\times {ℝ}^d\times \cdots \times {ℝ}^d$ that is concave in the first variable, jointly convex in the last $N-1$ variables, and such that

$$\left(u_1\left(x\right),u_2\left(x\right),\dots ,u_{N-1}\left(x\right)\right)={\nabla }_{2,\dots ,N}H\left(x,x,\dots ,x\right)$$

for almost all $x\in \Omega$. Moreover, $H$ is $N$-antisymmetric in a sense made precise later, and also $N$-sub-antisymmetric in the sense that for all $X\in {\Omega }^N$ the sum ${\sum }_{i=0}^{N-1}H\left({\sigma }^i\left(X\right)\right)\le 0$ is nonpositive, $\sigma$ being the permutation that shifts the coordinates of $X$ leftward one slot and places the first coordinate last. This result 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 $\Omega$.

Keywords

$N$-cyclically monotone vector fields, Krauss theorem

Mathematical Subject Classification 2010

Primary: 49J40

Milestones

Received: 7 January 2013

Revised: 2 October 2013

Accepted: 7 October 2013

Published: 26 July 2014