The Aronsson-Euler equation for absolutely minimizing Lipschitz extensions with respect to Carnot-Carathéodory metrics

Abstract

Abstract. We derive the Euler-Lagrange equation (also known in this setting as the Aronsson-Euler equation) for absolute minimizers of thewhere Ω ⊂ G is an open subset of a Carnot group, ∇ 0 u denotes the horizontal gradient of u : Ω → R, and the Lipschitz class is defined in relation to the Carnot-Carathéodory metric. In particular, we show that absolute minimizers are infinite harmonic in the viscosity sense. As a corollary we obtain the uniqueness of absolute minimizers in a large class of groups. This result extends previous work of Jensen and of Crandall, Evans and Gariepy. We also derive the Aronsson-Euler equation for more "regular" absolutely minimizing Lipschitz extensions corresponding to those Carnot-Carathéodory metrics which are associated to "free" systems of vector fields.

Starshapedeness for fully non‐linear equations in Carnot groups

“…which establishes (39). (c) From (a) and (b) the desired conclusion f ⋆ ≤ f follows thanks to the assumption (9). Since f = f ⋆ in Ω, for any ℓ ∈ (0, 1) we have…”

“…Finally, the fully nonlinear operator (20) fulfills the hypothesis ( 5), ( 6), (7) (Lemma 2.2 and (15)), whereas (9) follows from the comparison principle established in [7], [9] and also in [70]. For a further extension the reader can see [71].…”

“…Yet another important model for Theorem is the horizontal ‐ Laplacean defined by Similarly to its Euclidean predecessor, the fully non‐linear operator is formally obtained as the limiting case as of the operator in . The existence and uniqueness of viscosity solutions for the horizontal ‐Laplacean in Carnot groups was proved in , and subsequently in for general Hörmander type vector fields, see also . A more probabilistic approach to existence and uniqueness of viscosity solutions for is obtained from the theory of tug‐of‐war games.…”

The Aronsson Equation, Lyapunov Functions, and Local Lipschitz Regularity of the Minimum Time Function

“…For results for equation (AE) especially in the x dependent case, we also refer to the paper by the author [28] and the references therein, see also [27,26]. In particular we mention that equation (AE) has been studied in Carnot groups by Bieske-Capogna [9], by Bieske [8] in the Grushin space, and by Wang [30] in the case of C 2 and homogeneous Hamiltonians with a Carnot Caratheodory structure.…”

Asymptotic mean value formula for sub- p -harmonic functions on the Heisenberg group

“…Before proceeding, we would like to mention some motivations related to our research. Since Hörmanders work [22] the study of partial differential equations of sub-elliptic type like (1.6), (1.8) and (1.10) has received a strong impulse, see, e.g., [3], [4], [7], [11], [12], [13], [17], [31], [32] etc. These equations arise in many different settings: geometric theory of several complex variables, curvature problems for CR-manifolds, sub-Riemannian geometry, diffusion processes, control theory, human vision; see, e.g., [9], [20].…”