I have been reading Evan's book on PDE's , namely the section on calculus of variations. The author then gives conditions so that we can have existence and uniqueness of a minimizer. Then later we have a notion of weak solution of the Euler-Lagrange equation , and then one can give conditions so that a minimizer is also a weak solution of the PDE. Then he gives another condition so that we have that a weak solution of the PDE is a minimizer.
Now if we consider all of these conditions together, I belive we obtain a existence and uniqueness theorem for the euler-lagrange equations supposing that $L$ satisfies all of them. In theory this seems great, in practive I was wondering which sort of "interesting" lagrangians will satisfy all of these assumptions. Namely is there any interesting PDE which can be seen as the euler-lagrange equation of a suitable lagrangian and for which $L$ satisfies all of the conditions so that we can obtain a uniqueness and existence result just from calculus of variations ?
Any insight is appreciated, thanks in advance.