Regularity results for quasilinear degenerate elliptic obstacle problems in Carnot groups

Let {X-1 ,..., X-m} be a basis of the space of horizontal vector fields on the Carnot group G = (R-N, circle)(m < N). We establish regularity results for solutions to the following quasilinear degenerate elliptic obstacle problem integral(Omega) << AXu , Xu > (p-2/2) AXu, X( v - u )> dx >= integral(Omega) B(x, u, Xu)(v - u)dx + integral(Omega) < f(x), X(v - u)> dx, for all v is an element of K-psi(theta)(Omega), where A = (aij (x))(mxm) is a symmetric positive-definite matrix with measurable coefficients, p is close to 2, K-psi(theta)(Omega) = { V is an element of HW1.P(Omega): v >= psi a.e. in Omega, v - theta is an element of HWO1.P (Omega)}, psi is a given obstacle function theta is a boundary value function with theta >= psi. We first prove the C-X(O,alpha) regularity of solutions provided that the coefficients of A are of vanishing mean oscillation (VMO). Then the C-X(1,alpha) a regularity of solutions is obtained if the coefficients belong to the class BMO omega which is a proper subset of VMO.