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

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.

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References in corpus (1)

• Anti-selfdual Lagrangians II: Unbounded non self-adjoint operators and evolution equations

Cited by in corpus (4)

• On the existence of Hamiltonian paths connecting Lagrangian submanifolds
• Anti-selfdual Lagrangians II: Unbounded non self-adjoint operators and evolution equations
• Selfdual Variational Principles for Periodic Solutions of Hamiltonian and Other Dynamical Systems
• Anti-selfdual Hamiltonians: Variational resolutions for Navier-Stokes and other nonlinear evolutions