= Solution
A minimizing sequence in $\operatorname{Lip}_g(\Omega,k)$ is uniformly bounded and equi-Lipschitz. The <Arzela-Ascoli theorem> gives a uniformly convergent subsequence with limit $u$ having the same boundary data and Lipschitz constant at most $k$. Its gradients have a weak-star convergent subsequence in $L^\infty$, with limit $Du$. Convexity of $F$ makes the integral functional weak-star lower semicontinuous, so
$$
\mathcal F[u]\leq\liminf_j\mathcal F[u_j].
$$
Thus $u$ attains the infimum by the <direct method in the calculus of variations>.
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