# Uniqueness of minimizers in the Calculus of Variations

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.

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 $$u(x,y)=\min\{1 - x, 1 - y\}.$$