[Une théorie des lagrangiens anti-autoduaux : cas stationnaire]
On introduit et développe la notion de Lagrangien anti-autodual qui apparait dans plusieurs problèmes de géométrie et de physique théorique. Cette classe inclut les champs de gradient de fonctions convexes qui sont à la base de systèmes dissipatifs, mais aussi contient les opérateurs antisymétriques qui, par contre, engendrent des flots conservatifs. Comme pour les équations autoduales de Yang–Mills, Seiberg–Witten et Ginzburg–Landau, ces lagrangiens permettent la résolution variationnelle de plusieurs équations différentielles du premier ordre qui ne rentrent pas donc dans le cadre de la théorie de Euler–Lagrange.
We develop the concept of anti-self dual Lagrangians that seems inherent to many problems in mathematical physics, Riemannian geometry, and differential equations. On one hand, they represent gradients of convex functions which usually drive dissipative systems, and on the other, their structure is rich enough to also cover – certain representations of – skew-symmetric operators which normally generate unitary flows. These Lagrangians provide variational formulations and resolutions for several non-potential boundary value problems many of which do not fit in the Euler–Lagrange theory. Solutions are minima of functionals of the form where L is an anti-self dual Lagrangian and where Λ is a skew-adjoint operator. However, and just like the self (antiself) dual equations of quantum field theory (e.g. Yang–Mills) the equations associated to minimal solutions of our variational problems are not derived from the fact they are critical points of the associated functionals, but because they are also zeroes of the corresponding Lagrangians.
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@article{CRMATH_2005__340_3_245_0,
author = {Nassif Ghoussoub},
title = {A theory of anti-selfdual {Lagrangians:} stationary case},
journal = {Comptes Rendus. Math\'ematique},
pages = {245--250},
publisher = {Elsevier},
volume = {340},
number = {3},
year = {2005},
doi = {10.1016/j.crma.2004.12.010},
language = {en},
}
Nassif Ghoussoub. A theory of anti-selfdual Lagrangians: stationary case. Comptes Rendus. Mathématique, Volume 340 (2005) no. 3, pp. 245-250. doi : 10.1016/j.crma.2004.12.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.12.010/
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