# When does a PDE solve a variational problem?

I understand that for a functional $J[f]$ on the space of differentiable functions $f$ on some domain, setting $\delta J[f]|_{f=f_0} = 0$ yields a (possibly nonlinear) partial differential equation in $f_0$, the $f$ that minimizes $J$. Solutions are non-degenerate if extrema of $J$ are locally unique, and the uniqueness of the PDE is related to the number of extrema of $J$.

My question is if, given a known (nonlinear) partial differential equation with solutions $u$, there is a general method to construct a functional that the $u$ will minimize?

I am not interested in functionals of the form $(f-u)^2$ and other functionals that contain the explicit form of the solutions $u$, and would like to know if there is a method to find $J$ given the terms in the PDE. I'm aware that this can be done for all linear equations such as $Lu=v$ in a straightforward way; namely, for $L\cdot u=v$, $J[u] = \langle u\cdot L\cdot u \rangle - \langle v \cdot u \rangle - \langle u \cdot v \rangle$; but don't know of any extensions to nonlinear PDEs.

• This is definitely not easy. Have a look at chapter 8.2 (Existence of minimizers) of Evans' Partial Differential Equations. Jun 1, 2013 at 16:45
• Answers are essentially going to be duplicates of physics.stackexchange.com/q/20298/2451 and links therein. Jun 1, 2013 at 17:21