Regularity in the obstacle problem for parabolic non-divergence operators of Hörmander type

Abstract

In this paper we continue the study initiated in [15] concerning the obstacle problem for a class of parabolic non-divergence operators structured on a set of vector fields X={X1,…,Xq} in Rn with C∞-coefficients satisfying Hörmanderʼs finite rank condition, i.e., the rank of Lie[X1,…,Xq] equals n at every point in Rn. In [15] 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 [15] we in this paper assume, in addition, that there exists a homogeneous Lie group G=(Rn,∘,δλ) such that X1,…,Xq are left translation invariant on G and such that X1,…,Xq are δλ-homogeneous of degree one.