Let $f \colon \mathbb R^2 \to \mathbb R$ be the function defined by 
$$
f(x,y):= (x^+)^2 + (y^+)^2
$$
where $a^+ = \max\{a,0\}$ for any real number $a$. 


Given a Lipschitz regular domain $\Omega \subset \mathbb R^2$ and a "nice" function $\varphi \colon \partial \Omega \to \mathbb R$ I want to study the problem 
$$
\min \left\{ \int\_\Omega f(Du) \, dx :\, u \in \text{Lip}(\Omega) \text{ and } u = \varphi \text{ on } \partial \Omega \right\}. 
$$


Existence of minimizers should not be a severe issue and should follow easily (and classically) from the convexity of $f$ 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 *not* strictly convex, so I do not see a way to discuss uniqueness of minimizers. Any help? Thanks. 



Uniqueness of minimizers in the Calculus of Variations
There's no uniqueness in general. If $\Omega$ is a unit square, and $$\varphi(x, y) = (1-x-y)^+,$$ then obviously any $u$ which is decreasing with respect to both $x$ and $y$, and which matches the boundary condition, has energy zero, and hence it is a minimiser. However, there are [many $u$ like that](https://www.wolframalpha.com/input/?i=Plot3d%5B%7BMax%5B1-x-y,0%5D,Min%5B1-x,1-y%5D%7D,%7Bx,0,1%7D,%7By,0,1%7D%5D), for example, $$u(x,y)=(1-x-y)^+$$ or $$u(x,y)=\min\{1 - x, 1 - y\}.$$


