A variational theory for monotone vector fields

TL;DR

This paper introduces a variational framework for monotone vector fields using selfdual Lagrangians, enabling the analysis and solution of PDEs driven by maximal monotone operators through convex minimization.

Contribution

It develops a novel variational approach using selfdual Lagrangians to analyze and invert maximal monotone vector fields in Banach spaces.

Findings

Associates a convex selfdual Lagrangian to each monotone vector field

Reduces inversion of vector fields to convex minimization problems

Enables application of convex analysis methods to monotone operators

Abstract

Monotone vector fields were introduced almost 40 years ago as nonlinear extensions of positive definite linear operators, but also as natural extensions of gradients of convex potentials. These vector fields are not always derived from potentials in the classical sense, and as such they are not always amenable to the standard methods of the calculus of variations. We describe here how the selfdual variational calculus developed recently by the author, provides a variational approach to PDEs and evolution equations driven by maximal monotone operators. To any such a vector field T on a reflexive Banach space X, one can associate a convex selfdual Lagrangian L_T on phase space X x X* that can be seen as a "potential" for T, in the sense that the problem of inverting T reduces to minimizing the convex energy L_T. This variational approach to maximal monotone operators allows their theory to be analyzed with the full range of methods --computational or not-- that are available for variational settings. Standard convex analysis (on phase space) can then be used to establish many old and new results concerned with the identification, superposition, and resolution of such vector fields.