Preface

How to solve partial differential systems by completing the square. This could well have been the title of this monograph as it grew into a project to develop a systematic approach for associating suitable nonnegative energy functionals to a large class of partial differential equations (PDEs) and evolutionary systems. The minima of these functionals are to be the solutions we seek, not because they are critical points (i.e., from the corresponding Euler-Lagrange equations) but from also being zeros of these functionals. The approach can be traced back to Bogomolnyi’s trick of “completing squares” in the basic equations of quantum field theory (e.g., Yang-Mills, Seiberg-Witten, Ginzburg-Landau, etc.,), which allows for the derivation of the so-called self (or antiself) dual version of these equations. In reality, the “self-dual Lagrangians” we consider here were inspired by a variational approach proposed – over 30 years ago – by Brézis and Ekeland for the heat equation and other gradient flows of convex energies. It is based on Fenchel-Legendre duality and can be used on any convex functional – not just quadratic ones – making them applicable in a wide range of problems. In retrospect, we realized that the “energy identities” satisfied by Leray’s solutions for the Navier-Stokes equations are also another manifestation of the concept of self-duality in the context of evolution equations. The book could have also been entitled How to solve nonlinear PDEs via convex analysis on phase space. Indeed, the self-dual vector fields we introduce and study here are natural extensions of gradients of convex energies – and hence of selfadjoint positive operators – which usually drive dissipative systems but also provide representations for the superposition of such gradients with skew-symmetric operators, which normally generate conservative flows. Most remarkable is the fact that self-dual vector fields turned out to coincide with maximal monotone operators, themselves being far-reaching extensions of subdifferentials of convex potentials. This means that we have a one-to-one correspondence between three fundamental notions of modern nonlinear analysis: maximal monotone operators, semigroups of contractions, and self-dual Lagrangians. As such, a large part of nonlinear analysis can now be reduced to classical convex analysis on phase space, with self-dual Lagrangians playing the role of potentials for monotone vector fields according to