= Comparison principle for convex variational integrals
One precise <comparison principle for convex variational integrals> concerns minimizers of $E[u]=\int_\Omega F(\nabla u)dx$ on a bounded Lipschitz domain. Let $1<p<\infty$, take differentiable strictly convex $F$ with $|DF(\xi)|\le C(1+|\xi|^{p-1})$, and let $u,v\in W^{1,p}$ be weak stationary solutions whose boundary traces satisfy $u\le v$. Then $u\le v$ almost everywhere in the domain.
Use $w=(u-v)^+$, the <positive part> of the difference. Its zero trace makes it an admissible test in the two weak <Euler Lagrange equations>. Subtracting gives
$$
0=\int_{\{u>v\}}[DF(\nabla u)-DF(\nabla v)]\cdot(\nabla u-\nabla v)dx.
$$
Strict convexity makes the integrand positive unless the two gradients agree. Hence $\nabla w=0$ almost everywhere, and the zero-trace <Poincaré inequality> forces $w=0$. The growth bound makes these test pairings integrable.
This proves why ordered affine solutions can bound a minimizer in the <bounded slope condition> argument. Convexity without further strictness or uniqueness is insufficient to order every pair of minimizers: if $F=0$, every admissible function minimizes the energy and two equal-boundary competitors can cross.
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