= Solution
For $y\in\partial\Omega$, let $b_y^-\leq g\leq b_y^+$ 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 for convex variational integrals>[comparison principle] gives
$$
b_y^-(x)\leq u(x)\leq b_y^+(x).
$$
Since both barriers equal $u(y)$ at $y$ and are $K$-Lipschitz,
$$
|u(x)-u(y)|\leq K|x-y|
$$
for $x\in\Omega$ and $y\in\partial\Omega$. The supplied boundary-to-interior criterion now gives $\operatorname{Lip}(u)\leq K$.
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