Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 45, 2020, 293-303

HIGHER INTEGRABILITY OF MINIMIZERS OF DEGENERATE FUNCTIONALS IN CARNOT–CARATHÉODORY SPACES

Patrizia Di Gironimo and Flavia Giannetti

Università degli Studi di Salerno, Dipartimento di Matematica
Via Giovanni Paolo II 132, 84084 Fisciano (SA), Italia; pdigironimo 'at' unisa.it

Università degli Studi di Napoli, Dipartimento di Matematica e Applicazioni "R. Caccioppoli"
Via Cintia, 80126 Napoli, Italia; giannett 'at' unina.it

Abstract. In this paper, we prove a higher integrability result for the horizontal gradient of a minimizer of a functional of the type

I(Ω,u) = ∫_Ω∑_i,ja_i,jX_iuX_ju dx

whose matrix of the coefficients A(x) = ^tA(x) satisfies the anisotropic bounds |ξ|²/K(x) ≤ ⟨A(x)ξ,ξ⟩ ≤ K(x)|ξ|² ∀ξ ∈ Rⁿ, for a.e. x ∈ Ω,

where the ellipticity function K(x) ∈ A₂ ∩ RH_τ, τ opportunely related to the homogeneous dimension, and is such that the pair (K,1/K) ∈ A₁.

2010 Mathematics Subject Classification: Primary 49N60.

Key words: Hörmander vector fields, A_p weights, regularity.

Reference to this article: P. Di Gironimo and F. Giannetti: Higher integrability of minimizers of degenerate functionals in Carnot–Carathéodory spaces. Ann. Acad. Sci. Fenn. Math. 45 (2020), 293-303.

Full document as PDF file

https://doi.org/10.5186/aasfm.2020.4509