Regularity results for quasilinear degenerate elliptic obstacle problems in Carnot groups

Abstract

Let $\{X_1,\dots ,X_m\}$ be a basis of the space of horizontal vector fields on the Carnot group $\mathbb{G}=({\mathbb{R}}^N,\circ )(m<N)$. We establish regularity results for solutions to the following quasilinear degenerate elliptic obstacle problem

where $A=(a_{ij}(x){)}_{m\times m}$ is a symmetric positive-definite matrix with measurable coefficients, $p$ is close to 2, ${\mathcal{K}}_{\psi }^{\theta }(\Omega )=\{v\in H{W}^{1,p}(\Omega ):v\ge \psi \text{a.e.}\mathrm{in}\Omega ,v-\theta \in H{W}_0^{1,p}(\Omega )\}$, $\psi$ is a given obstacle function, $\theta$ is a boundary value function with $\theta \ge \psi$ . We first prove the $C_X^{0,\alpha }$ regularity of solutions provided that the coefficients of $A$ are of vanishing mean oscillation (VMO). Then the $C_X^{1,\alpha }$ regularity of solutions is obtained if the coefficients belong to the class ${\text{BMO}}_{\omega }$ which is a proper subset of VMO.