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Boundary data $g$ satisfy the bounded slope condition with constant $K$ if, at every boundary point $x_0$, there are affine functions ${\ell }_{x_0}^{-}$ and ${\ell }_{x_0}^{+}$ such thatand $|\nabla {\ell }_{x_0}^{\pm }|\le K$. Both the touching condition and the common slope bound matter; arbitrary affine upper/lower bounds would be much weaker.

An affine datum $g(x)=a\cdot x+b$ satisfies the condition with $K=|a|$, using that same affine function for both supports. For a suitable convex variational problem on a convex domain, these supports serve as barriers through the comparison principle for convex variational integrals. Translation comparisons then produce a global Lipschitz continuity bound. This consequence requires the structural and boundary hypotheses of the variational problem, rather than following from the boundary inequalities alone.

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.

For $y\in \partial \Omega$, let $b_y^{-}\le g\le b_y^{+}$ be the affine barriers from the bounded slope condition. Their constant gradients satisfy the Euler-Lagrange equation, so they minimize the autonomous convex functional for their own boundary values. The comparison principle givesSince both barriers equal $u(y)$ at $y$ and are $K$-Lipschitz,for $x\in \Omega$ and $y\in \partial \Omega$. The supplied boundary-to-interior criterion now gives $\mathrm{Lip}(u)\le K$.