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Anti-selfdual Lagrangians: Variational resolutions of non self-adjoint equations and dissipative evolutions

Nassif Ghoussoub

TL;DR

The paper addresses solving non-potential boundary value problems and dissipative parabolic evolutions by introducing and developing the calculus of anti-self dual (ASD) Lagrangians, which extend gradients of convex functions and accommodate skew-symmetric components. The solutions are minima of functionals $I(u)=L(u,\Lambda u)$ or $I(u)=\int_{0}^{T} L(t, u(t), \dot u(t) + \Lambda_{t}u(t))\,dt$, and crucially correspond to zeros of the Lagrangian $L$ rather than critical points of $I$. They define ASD and $R$-antiselfdual Lagrangians and show how many operator equations, including maximal monotone operators and anti-Hamiltonian systems, admit variational representations via $I$ attaining its minimum at zero, thus yielding existence and uniqueness results. The framework is extended with boundary Lagrangians and boundary operators, lifted to path spaces for parabolic problems and gradient flows, and complemented by autonomous/time-dependent variants that generate contraction semigroups and energy identities, thereby unifying dissipative dynamics within a variational setting.

Abstract

We develop the concept and the calculus of anti-self dual (ASD) Lagrangians which seems inherent to many questions in mathematical physics, geometry, and differential equations. They are natural extensions of gradients of convex functions --hence of self-adjoint positive operators-- which usually drive dissipative systems, but also rich enough to provide representations for the superposition of such gradients with skew-symmetric operators which normally generate unitary flows. They yield variational formulations and resolutions for large classes of non-potential boundary value problems and initial-value parabolic equations. Solutions are minima of functionals of the form $I(u)=L(u, \Lambda u)$ (resp. $I(u)=\int_{0}^{T}L(t, u(t), \dot u (t)+\Lambda_{t}u(t))dt$) where $L$ is an anti-self dual Lagrangian and where $\Lambda_{t}$ are essentially skew-adjoint operators. However, and just like the self (and antiself) dual equations of quantum field theory (e.g. Yang-Mills) the equations associated to such minima are not derived from the fact they are critical points of the functional $I$, but because they are also zeroes of the Lagrangian $L$ itself.

Anti-selfdual Lagrangians: Variational resolutions of non self-adjoint equations and dissipative evolutions

TL;DR

The paper addresses solving non-potential boundary value problems and dissipative parabolic evolutions by introducing and developing the calculus of anti-self dual (ASD) Lagrangians, which extend gradients of convex functions and accommodate skew-symmetric components. The solutions are minima of functionals or , and crucially correspond to zeros of the Lagrangian rather than critical points of . They define ASD and -antiselfdual Lagrangians and show how many operator equations, including maximal monotone operators and anti-Hamiltonian systems, admit variational representations via attaining its minimum at zero, thus yielding existence and uniqueness results. The framework is extended with boundary Lagrangians and boundary operators, lifted to path spaces for parabolic problems and gradient flows, and complemented by autonomous/time-dependent variants that generate contraction semigroups and energy identities, thereby unifying dissipative dynamics within a variational setting.

Abstract

We develop the concept and the calculus of anti-self dual (ASD) Lagrangians which seems inherent to many questions in mathematical physics, geometry, and differential equations. They are natural extensions of gradients of convex functions --hence of self-adjoint positive operators-- which usually drive dissipative systems, but also rich enough to provide representations for the superposition of such gradients with skew-symmetric operators which normally generate unitary flows. They yield variational formulations and resolutions for large classes of non-potential boundary value problems and initial-value parabolic equations. Solutions are minima of functionals of the form (resp. ) where is an anti-self dual Lagrangian and where are essentially skew-adjoint operators. However, and just like the self (and antiself) dual equations of quantum field theory (e.g. Yang-Mills) the equations associated to such minima are not derived from the fact they are critical points of the functional , but because they are also zeroes of the Lagrangian itself.

Paper Structure

This paper contains 21 sections, 40 theorems, 288 equations.

Table of Contents

  1. Introduction
  2. Basic variational properties of R-antiselfdual Lagrangians
  3. Permanence properties of Anti-selfdual Lagrangians
  4. ASD Lagrangians in variational problems with no boundary constraint
  5. ASD Lagrangians as representations of certain maximal monotone operators
  6. ASD Lagrangians and anti-Hamiltonian systems
  7. ASD Lagrangians associated to boundary value problems
  8. Anti-selfduality involving boundary Lagrangians
  9. Variational properties of ASD Lagrangians with boundary terms
  10. Variational principle for operators which are positive modulo a boundary
  11. ASD Lagrangians on intermediate Hilbert spaces
  12. ASD Lagrangians for coupled equations with prescribed boundaries
  13. Time dependent anti-self dual Lagrangians
  14. ASD Lagrangians on path spaces
  15. ASD Lagrangians in the calculus of variations
  16. ...and 6 more sections

Key Result

Proposition 2.1

Let L be a convex lower-semi continuous functional on a reflexive Banach space X\times X^{*}. Assume that L is a partially R-Antiselfdual Lagrangian and that for some x_{0}\in X, the function p\to L(x_{0},p) is bounded above on a neighborhood of the origin in X^{*}. Then there exists \bar{x}\in X, s

Theorems & Definitions (57)

  • Definition 2.1
  • Proposition 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Lemma 3.1
  • Proposition 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • ...and 47 more