Abstract

We present a uniqueness result of uniformly continuous solutions for a general minimization problem in the Calculus of Variations. We minimize the functional Iλ(u):=∫Ωφ(∇u)+λu\mathcal{I}_\lambda(u):=\int_\Omega \varphi(\nabla u) +\lambda u with φ\varphi a convex but not necessarily strictly convex function, Ω\Omega an open set of RN\mathbb{R}^N with N∈NN\in \mathbb{N} and λ∈R\lambda\in\mathbb{R}. The proof is based on the two following main points: the functional Iλ\mathcal{I}_\lambda is invariant under translations and we assume that the function φ\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 Iλ\mathcal{I}_\lambda that are presented in the article.

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Keywords

  • Calculus of variations
  • translation invariance
  • non strictly-convex function
  • uniqueness

Mathematics Subject Classification

35A02, 49N99

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Suggested citation

B. Lledos. “A Uniqueness Result for a Translation Invariant Problem in the Calculus of Variations.” Journal of Convex Analysis 31 (2024), No. 1, 121–130. https://doi.org/10.68381/jca31008

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Published by Heldermann Verlag, 2024. Rights now held by Banach Press.