A Uniqueness Result for a Translation Invariant Problem in the Calculus of Variations

We present a uniqueness result of uniformly continuous solutions for a general minimization problem in the Calculus of Variations. We minimize the functional $\mathcal{I}_\lambda(u):=\int_\Omega \varphi(\nabla u) +\lambda u$ with $\varphi$ a convex but not necessarily strictly convex function, $\Omega$ an open set of $\mathbb{R}^N$ with $N\in \mathbb{N}$ and $\lambda\in\mathbb{R}$. The proof is based on the two following main points: the functional $\mathcal{I}_\lambda$ is invariant under translations and we assume that the function $\varphi$ is not affine on any non-empty open set. This provides a shorter proof and/or an extension for some already known uniqueness results for functionals of the type $\mathcal{I}_\lambda$ that are presented in the article.