Global higher integrability for very weak solutions to nonlinear subelliptic equations

Abstract

Abstract In this paper we consider the following nonlinear subelliptic Dirichlet problem: { X ∗ A ( x , u , X u ) + B ( x , u , X u ) = 0 , x ∈ Ω , u − u 0 ∈ W X , 0 1 , r ( Ω ) , $$ \textstyle\begin{cases} X^{*}A(x,u,Xu)+ B(x,u,Xu)=0,& x\in\Omega,\\ u-u_{0}\in W_{X,0}^{1,r}(\Omega), \end{cases} $$ where X = { X 1 , … , X m } $X=\{X_{1},\ldots,X_{m}\}$ is a system of smooth vector fields defined in R n $\mathbf{R}^{n}$ with globally Lipschitz coefficients satisfying Hörmander’s condition, and we prove the global higher integrability for the very weak solutions.

Keywords

• nonlinear subelliptic equations
• very weak solutions
• global higher integrability