This is something of a sanity check as I don't have that much formal background in the calculus of variations. My intuition to the block-quoted question below is that the answer is affirmative but I'm afraid I'm making a big blunder.
I am trying to understand the role of boundary conditions in deriving the Euler-Lagrange equations, and in particular whether they are necessary at all. To make things precise, assume $$ R : X \to \mathbb{R} $$ Is a convex Gateaux-differtiable functional. $X$ is a Hilbert space of functions, like say a closed subspace of the Sobolev space $H^p$. $R$ is furthermore of the form: $$ R(u)= \int_\Omega L(x,u) dx $$ for some region $\Omega$ of $\mathbb{R}^n$. $L$ is nice (satisfying all the usual requirements on coercivity, smoothness and so on) and is allowed to depend on $u\in X$. The case where $L$ depends on the derivative and higher order derivatives up to order $p$ of $u$ is also of interest. However, I should stress that the "base case" without derivatives is of independent interest to me.
My question is the following:
Does there exist a (nonlinear) partial differential operator $A$ (Euler-Lagrange?) such that the unique minimizer $u$ of $R$ is the unique solution to $A(u)=0$.
What this boils down to is that I am trying to ascertain the role of boundary conditions in the derivation of the Euler-Lagrange equations as presented in for instance Evans and whether it is possible to charactarize optimization problem by a certain partial differential operator (and in this application assuming boundary conditions makes no sense).
As a remark, my confusion about whether the Euler-Lagrange equations depend on boundary conditions or not probably comes from me only studying evans, where the boundary conditions are naturally part of the problem, as the end goal is to study PDE. Here on the other hand, the objectives are switched and the end goal is to study optimization using PDE.