Regularity results for quasilinear degenerate elliptic obstacle problems in Carnot groups

Abstract

Let $\{X_1 ,\ldots ,X_m \}$ 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 \begin{align*} &\int\limits_\Omega\! {\langle \langle AXu,Xu\rangle ^{\frac{p-2}{2}}AXu,X(v-u)\rangle } dx\\ &\quad\geq\int\limits_\Omega\! {B(x,u,Xu)(v-u)}dx +\int\limits_\Omega \!{\langle f(x),X(v-u)\rangle } dx,\quad \text{for all } v\in \mathcal{K}_\psi^\theta (\Omega), \end{align*} where $A=(a_{ij}(x))_{m\times m}$ is a symmetric positive-definite matrix with measurable coefficients, $p$ is close to 2, $\mathcal{K}_\psi ^\theta(\Omega )=\{v\in HW^{1,p}(\Omega)\colon v\geq \psi \;\mbox{a.e.}\;{\rm in}\;\Omega ,v-\theta\in HW_0^{1,p} (\Omega )\}$, $\psi$ is a given obstacle function, $\theta$ is a boundary value function with $\theta \geq \psi$ . 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 $\mbox{BMO}_\omega$ which is a proper subset of VMO.