Integral representation of local functionals depending on vector fields

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.

G-convergence of elliptic and parabolic operators depending on vector fields

“…Theorem 3.3. Consider a sequence (a h ) h ⊂ M Ω (α, β, p) and the related sequence of elliptic operators (A h ) h , defined in (7).…”

“…Before entering into the details, we want to recall that the literature concerning homogenization, G-convergence and integral representation of abstract functionals depending on vector fields is pretty vast, ranging from more rigid structures like Carnot groups, see e.g. [1, 10-12, 16, 19] and the references therein, up to the more general setting considered in [7,8,17,18]. The setting we will take into account embraces the huge family of vector fields satisfying the Hörmander condition.…”

The Aronsson equation for absolute minimizers of supremal functionals in Carnot–Carathéodory spaces

“…We stress that this point of view is pretty general and encompasses, among other things, the Euclidean setting and many interesting sub-Riemannian manifolds. On the other hand, its rich analytical structure allows to study many interesting problems in great generality (see, e.g., [17,[25][26][27] and references therein).…”

The Aronsson Equation for Absolute Minimizers of Supremal Functionals in Carnot-Carathéodory Spaces

“…We stress that this point of view is pretty general and encompasses, among other things, the Euclidean setting and many interesting sub-Riemannian manifolds. On the other hand its rich analytical structure allows to study many interesting problems in great generality (see for example [EPV,MSC,MPSC,MPSC2] and references therein).…”