# Regularity results for quasilinear degenerate elliptic obstacle problems in Carnot groups

### Guangwei Du

Northwestern Polytechnical University, Xi'an, Shaanxi, China### Pengcheng Niu

Northwestern Polytechnical University, Xi'an, Shaanxi, China### Junqiang Han

Northwestern Polytechnical University, Xi'an, Shaanxi, China

## Abstract

Let ${X_{1},…,X_{m}}$ be a basis of the space of horizontal vector fields on the Carnot group $G=(R_{N},∘)(m<N)$. We establish regularity results for solutions to the following quasilinear degenerate elliptic obstacle problem

where $A=(a_{ij}(x))_{m×m}$ is a symmetric positive-definite matrix with measurable coefficients, $p$ is close to 2, $K_{ψ}(Ω)={v∈HW_{1,p}(Ω):v≥ψa.e.inΩ,v−θ∈HW_{0}(Ω)}$, $ψ$ is a given obstacle function, $θ$ is a boundary value function with $θ≥ψ$ . We first prove the $C_{X}$ regularity of solutions provided that the coefficients of $A$ are of vanishing mean oscillation (VMO). Then the $C_{X}$ regularity of solutions is obtained if the coefficients belong to the class $BMO_{ω}$ which is a proper subset of VMO.

## Cite this article

Guangwei Du, Pengcheng Niu, Junqiang Han, Regularity results for quasilinear degenerate elliptic obstacle problems in Carnot groups. Rend. Sem. Mat. Univ. Padova 141 (2019), pp. 65–105

DOI 10.4171/RSMUP/15