DOI: 10.68381/jca31008 ISSN: 0944-6532
A Uniqueness Result for a Translation Invariant Problem in the Calculus of Variations
Benjamin Lledos
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
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.