Uniqueness of minimizers in a problem in the Calculus of Variations - Part II 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.
 A: 
Existence of minimizers should not be a severe issue...

The proof of the existence follows form the Arzela-Ascoli theorem, but the proof is not entirely obvious. This is Proposition 1.1 in:
E. Giusti, Direct methods in the calculus of variations. World Scientific Publishing Co., Inc., River Edge, NJ, 2003. 
The statement is as follows:

Theorem. Let $F$ be a convex function, and let $\Omega\subset\mathbb{R}^n$ be bouded and open. Let $\varphi$ be 
  Lipschitz continuous on $\partial\Omega$ with the Lipschitz constant
  $\leq k$. Then the functional $$ I(u)=\int_\Omega F(Du)\, dx $$
  attains minimum in the class of $k$-Lipschitz functions on $\Omega$
  that agrees with $\varphi$ on $\partial\Omega$.

A: Consider $n=1$, $A=[-1,1]$, $\Omega=(0,1)$. You want to minimize $\int_0^1 |u'(x)|\,dx$ subject to Dirichlet conditions, say $u(0)=0$ and $u(1)=1$. Then it is quite obvious that every monotone function is a minimizer, so uniqueness does not hold.
A: I note that the result quoted from Giusti's book a priori limits the Lipschitz constant to a specific value. Without this limitation, a minimizer may not exist. Consider, for instance, the annulus $1<r<2$ in two dimensions, and the minimization of $\int_\Omega |\nabla u|$, with boundary condition u=r. We need to consider only radial functions, so the problem boils down to minimizing $\int_1^2 2\pi r |v'(r)|\,dr$ over all functions $v(r)$ with $v(1)=1$ and $v(2)=2$. If we require $v$ to have Lipschitz constant 1, the minimum is $3\pi$, achieved for $v(r)=r$. If we allow $v$ to have  any Lipschitz constant, the infimum is $2\pi$, approximated by functions localized near $r=1$, but not achieved.
