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.



Uniqueness of minimizers in a problem in the Calculus of Variations - Part II

> 
> *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 $\partial\Omega$.
> 
> 
> 


