🚀 go-pugleaf

A student here at UCR, Mary Legner, was trying to extend some theorems
about elliptic PDE of the form

Laplacian u = f(u)

to those of the form

Laplacian u = f(u,Du)

where Du is the gradient.  (Actually she is doing much fancier stuff,
but I'll keep it simple here.)  She was trying to find a variational
principle good for the latter kind.  Is there any functional of u and
Du that when extremized gives you the latter PDE?

I told her I was dubious becuase 1) if there was, some of the papers
she'd read would have mentioned it, and 2) in mechanics, there doesn't
seem to be any Lagrangian that gives the equations of motion

x'' = -x'

(which describes a particle with friction).  

What a drag that'd be if variational methods weren't applicable to 2nd
order elliptic ODE with first-derivative terms!