CMSMITACS Joint Conference 2007May 31  June 3, 2007
Delta Hotel, Winnipeg, Manitoba

How to solve PDEs by completing squares? Motivated in part by the basic equations of quantum field theory (e.g. YangMills, GinzburgLandau, etc....), we identify a large class of partial differential equations that can be solved via a newly devised "selfdual" variational calculus.
In both stationary and dynamic cases, such selfdual equations are not derived from the fact they are critical points of action functionals, but because they are also zeroes of appropriately chosen nonnegative Lagrangians. The class contains many of the basic families of linear and nonlinear, stationary and evolutionary partial differential equations: Transport equations, Nonlinear Laplace equations, CauchyRiemann systems, NavierStokes equations, but also infinite dimensional gradient flows of convex potentials (e.g. heat equations), nonlinear Schrödinger equations, Hamiltonian systems, and many other parabolicelliptic equations.