Consider the second-order elliptic boundary value problem
$$-(a(x) u(x)')' + b(x) u' + c(x)u = f(x)$$
over interval $I = [0,1]$, together with homogeneous Dirichlet boundary conditions $u(0)=u(1) = 0$.
For simplicity, we assume that $a$, $b$, and $c$ are differentiable and that $a$ satisfies the condition $a(x) \geq \alpha > 0$ over the interval.
There exists a unique solution $u$, but when that solution be characterized as the critical point of a convex energy functional? I am looking for conditions on the convection term. For example, when $b = 0$, then I can use the functional
$$ E(u) = \frac 1 2 \int_0^1 a(x) (u'(x))^2 + c(x) u(x)^2 dx. $$
That is well-known. But when $b \neq 0$, I can't see how to modify this functional to accommodate the convection term, even if $b$ is much smaller in magnitude than $a$ and $c$. I have not found this in any textbook either.