 Solution

Let $L=\mathrm{Lip}(u)<k$ and take $v\in {\mathrm{Lip}}_g(\Omega )$. For small $t>0$, $u_t=(1-t)u+tv$ has Lipschitz constant at most $(1-t)L+t\mathrm{Lip}(v)<k$. Constrained minimality and convexity giveCancellation gives $\mathcal{F}[u]\le \mathcal{F}[v]$, so $u$ is an unconstrained minimizer.