Take any convex set $A\subset\mathbb{R}^n$ which contains a neighborhood of the origin and let $f_A$ be the associated *Minkowski functional*
$$
f_A(x) = \inf\{\lambda>0\mid x\in\lambda A\},
$$
which turns to be a convex, homogeneous function defined on $\mathbb R^n$.
Given a Lipschitz regular domain $\Omega \subset \mathbb R^n$ and a "nice" function $\varphi \colon \partial \Omega \to \mathbb R$ I want to study the problem
$$
\min \left\{ \int_\Omega f_A(Du) \, dx :\, u \in \text{Lip}(\Omega) \text{ and } u = \varphi \text{ on } \partial \Omega \right\}.
$$

Existence of minimizers should not be a severe issue and should follow easily (and classically) from the convexity of $f_A$ and from the properties of $\varphi$ (like e.g. the bounded slope condition).

What about *uniqueness* of (Lipschitz) minimizers? It seems to me that this is quite a difficult task, as the function $f$ is *never* strictly convex, being 1-homogeneous. So I do not see a way to discuss uniqueness of minimizers, and actually I even doubt that uniqueness holds. Any help? Thanks.