= 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 that
$$
\ell^-_{x_0}(x)\le g(x)\le\ell^+_{x_0}(x)\quad(x\in\partial\Omega),\qquad
\ell^-_{x_0}(x_0)=g(x_0)=\ell^+_{x_0}(x_0),
$$
and $|\nabla\ell^\pm_{x_0}|\le K$. 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.
Back to article page
