Regularity results for quasilinear degenerate elliptic obstacle problems in Carnot groups

Du, Guangwei; Niu, Pengcheng; Han, Junqiang

Rendiconti del Seminario Matematico della Università di Padova, Tome 141 (2019), pp. 65-105.

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

$$\begin{array}{cc}& \underset{\Omega }{\int }\langle {\langle AXu,Xu\rangle }^{\frac{p-2}{2}}AXu,X(v-u)\rangle dx\hfill \\ & \ge \underset{\Omega }{\int }B(x,u,Xu)(v-u)dx+\underset{\Omega }{\int }\langle f\left(x\right),X(v-u)\rangle dx,\text{for}\text{all}v\in {𝒦}_{\psi }^{\theta }\left(\Omega \right),\hfill \end{array}$$

where $A={\left(a_{ij}\left(x\right)\right)}_{m\times m}$ is a symmetric positive-definite matrix with measurable coefficients, $p$ is close to 2, ${𝒦}_{\psi }^{\theta }\left(\Omega \right)=\{v\in H{W}^{1,p}\left(\Omega \right):v\ge \psi \text{a.e.}\mathrm{in}\Omega ,v-\theta \in H{W}_0^{1,p}\left(\Omega \right)\}$, $\psi$ 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.