Abstract
Given an open and bounded set
Ω
⊆
ℝ
n
{\Omega\subseteq\mathbb{R}^{n}}
and a family
𝐗
=
(
X
1
,
…
,
X
m
)
{\mathbf{X}=(X_{1},\ldots,X_{m})}
of Lipschitz vector fields on Ω, with
m
≤
n
{m\leq n}
, we characterize three classes of local functionals defined on first-order X-Sobolev spaces, which admit an integral representation in terms of X, i.e.
F
(
u
,
A
)
=
∫
A
f
(
x
,
u
(
x
)
,
X
u
(
x
)
)
𝑑
x
,
F(u,A)=\int_{A}f(x,u(x),Xu(x))\,dx,
with f being a Carathéodory integrand.
Integral representation and relaxation of local functionals on Cheeger–Sobolev spaces
