Interior [formula omitted] estimates for divergence degenerate elliptic systems in Carnot groups

Abstract

Let X1,…,Xq be the basis of the space of horizontal vector fields on a homogeneous Carnot group G=(Rn,∘)(q<n). We consider the following divergence degenerate elliptic system ∑β=1N∑i,j=1qXi(aαβij(x)Xjuβ)=∑i=1qXifαi,α=1,2,…,N where the coefficients aαβij are real valued bounded measurable functions defined in Ω⊂G, satisfying the strong Legendre condition and belonging to the space VMOloc(Ω) (defined by the Carnot–Carathéodory distance induced by the Xi’s). We prove interior HW1,p estimates (2≤p<∞) for weak solutions to the system.