
Journal of Convex Analysis 31 (2024), No. 1, 121130 Copyright Heldermann Verlag 2024 A Uniqueness Result for a Translation Invariant Problem in the Calculus of Variations Benjamin Lledos Institut de Mathématiques, CNRS UMR 5219, Université de Toulouse, France benjamin.lledos@math.univtoulouse.fr [Abstractpdf] 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 nonempty 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. Keywords: Calculus of variations, translation invariance, non strictlyconvex function, uniqueness. MSC: 35A02, 49N99. [ Fulltextpdf (120 KB)] for subscribers only. 