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 givesStrict 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.