= Solution
Part c supplies a <bounded slope condition> constant $K$. Choose $k>K$. Parts a and d give a constrained minimizer with $\operatorname{Lip}(u)\leq K<k$, and part b makes it a minimizer over all of $\operatorname{Lip}_g(B_1)$.
For $F(\xi)=\sqrt{1+|\xi|^2}$, the <Euler-Lagrange equation> is the <minimal surface equation for a graph>
$$
\operatorname{div}\left(\frac{Du}{\sqrt{1+|Du|^2}}\right)=0.
$$
The Lipschitz bound confines $Du$ to a compact set on which $D^2F$ is uniformly positive definite, so the equation is uniformly elliptic. Interior regularity gives $C^{1,\alpha}$ first, and repeated <Schauder estimate>[Schauder estimates] then give smoothness in the interior.
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