2005/03/24 by Nassif Ghoussoub, Ghoussoub, Nassif · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #math.AP
paper · pdf · doi:10.48550/arxiv.math/0503571
50 pages. For the most updated version of this paper, please visit http://www.pims.math.ca/~nassif/pims_papers.html
arxiv created 2005/03/24 · arxiv updated 2009/12/01
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, Λu) (resp. I(u)=∫0TL(t, u(t), u (t)+Λtu(t))dt) where L is an anti-self dual Lagrangian and where Λt 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.