A Uniqueness Result for a Translation Invariant Problem in the Calculus of Variations
Benjamin Lledos · Journal of convex analysis · 2024
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( abla u) +\lambda u I λ ( u ) : = ∫ Ω φ ( ∇ u ) + λ u with \varphi φ a convex but not necessarily strictly convex function, \Omega Ω an open set of \mathbb{R}^N R N with N\in \mathbb{N} N ∈ N and \lambda\in\mathbb{R} λ ∈ R . The proof is based on the two following main points: the functional \mathcal{I}_\lambda I λ is invariant under translations and we assume that the function \varphi φ 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 I λ that are presented in the article.