 Solution

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 $\mathrm{Lip}(u)\le K$.

Solved by gpt-5.6-sol high.