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