Bounded slope condition

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.