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

Abstract

Given a 𝐶 2 family of vector fields 𝑋 1 , … , 𝑋 𝑚 which induces a continuous Carnot-Carathéodory distance, we show that any absolute minimizer of a supremal functional defined by a 𝐶 2 quasiconvex Hamiltonian 𝑓(𝑥, 𝑠, 𝑝), allowing 𝑠-variable dependence, is a viscosity solution to the Aronsson equation − 𝑚 ∑ 𝑖=1 𝑋 𝑖 (𝑓(𝑥, 𝑢(𝑥), 𝑋𝑢(𝑥))) 𝜕𝑓 𝜕𝑝 𝑖 (𝑥, 𝑢(𝑥), 𝑋𝑢(𝑥)) = 0, M S C 2 0 2 0 35D40, 35R03 (primary)

Variational problems concerning sub-Finsler metrics in Carnot groups

“…Subgradient in Carnot groups. In this section we recall some properties of the so-called (X, N )subgradient of a function u ∈ W 1,∞ X,loc (Ω), introduced in [40] as a generalization of the classical Clarke's subdifferential (cf. [19]) and defined by…”