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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.

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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, for example, $$u(x,y)=(1-x-y)^+$$ or $$u(x,y)=\min\{1 - x, 1 - y\}.$$

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