Regularity in the Obstacle Problem for Parabolic Non-divergence Operators of H\"ormander type

Abstract

In this paper we continue the study initiated in [FGN] concerning the obstacle problem for a class of parabolic non-divergence operators structured on a set of vector fields X = {X_1,...,X_q} in R^n with C^1-coefficients satisfying H\"ormander's finite rank condition, i.e., the rank of Lie{X_1,...,X_q} equals n at every point in R^n. In [FGN] we proved, under appropriate assumptions on the operator and the obstacle, the existence and uniqueness of strong solutions to a general obstacle problem. The main result of this paper is that we establish further regularity, in the interior as well as at the initial state, of strong solutions. Compared to [FGN] we in this paper assume, in addition, that there exists a homogeneous Lie group G = (R^n, \circ, \delta_\lambda) such that {X_1,...,X_q} are left translation invariant on G and such that {X_1,...,X_q} are \delta_\lambda-homogeneous of degree one.