- Partial Lipschitz regularity of the minimum time function for sub-Riemannian control systems(arXiv)

Author : Paolo Albano, Vincenzo Basco, Piermarco Cannarsa

Abstract : In Euclidean space of dimension 2 or 3, we study a minimum time problem associated with a system of real-analytic vector fields satisfying Hörmander’s bracket generating condition, where the target is a nonempty closed set. We show that, in dimension 2, the minimum time function is locally Lipschitz continuous while, in dimension 3, it is Lipschitz continuous in the complement of a set of measure zero. In particular, in both cases, the minimum time function is a.e. differentiable on the complement of the target. In dimension 3, in general, there is no hope to have the same regularity result as in dimension 2. Indeed, examples are known where the minimum time function fails to be locally Lipschitz continuous.

2.Lipschitz regularity of minimizers of variational integrals with variable exponents (arXiv)

Author : Michela Eleuteri, Antonia Passarelli di Napoli

Abstract : In this paper we prove the Lipschitz regularity for local minimizers of convex variational integrals of the form

F(v,Ω)=∫ΩF(x,Dv(x))dx,

where, for n>2 and N≥1, Ω is a bounded open set in Rn, u∈W1,1(Ω,RN) and the energy density F:Ω×RN×n→R satisfies the so called variable growth conditions. The main novelty of the paper is that we assume an almost critical regularity in the Orlicz Sobolev setting for the energy density as a function of the x variable