反自对偶拉格朗日量：非自伴方程的变分解和耗散演化
Anti-selfdual Lagrangians: Variational resolutions of non self-adjoint equations and dissipative evolutions
摘要：
我们发展了反自对偶（ASD）拉格朗日量的概念和微积分，它似乎是数学物理、几何和微分方程中许多问题所固有的。它们是凸函数梯度的自然延伸--因此是自伴正算子的自然延伸--这通常驱动耗散系统，但也足够丰富，可以提供这种梯度与通常生成酉流的斜对称算子的叠加表示。他们产生变分制剂和决议的大类非潜在的边值问题和初值抛物方程。解是形式为$I(u)=L(u, Λu)$的泛函的极小值（分别为$I(u)=\int_{0}^{T}L(t, u(t), \dot u (t)+Λ_{t}u(t))dt$），其中$L$是反自对偶拉格朗日量，并且其中$Λ_{t}$本质上是斜伴随算子.然而，就像量子场论（例如杨-米尔斯）的自（和反自）对偶方程一样，与这种极小值相关的方程并不是因为它们是泛函$I$的临界点，而是因为它们也是拉格朗日量$L$本身的零点。
英文摘要：
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. $I(u)=\int_{0}^{T}L(t, u(t), \dot u (t)+Λ_{t}u(t))dt$) where $L$ is an anti-self dual Lagrangian and where $Λ_{t}$ 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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