The obstacle problem for parabolic non-divergence form operators of Hörmander type

Abstract

The main result established in this paper is the existence and uniqueness of strong solutions to the obstacle problem for a class of subelliptic operators in non-divergence form. The operators considered are structured on a set of smooth vector fields in R n , X = {X0, X1, ..., Xq}, q ≤ n, satisfying Hörmander's finite rank condition. In this setting, X0 is a lower order term while {X1, ..., X q} are building blocks of the subelliptic part of the operator. In order to prove this, we establish an embedding theorem under the assumption that the set {X0, X1, ..., Xq} generates a homogeneous Lie group. Furthermore, we prove that any strong solution belongs to a suitable class of Hölder continuous functions.

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

“…When we proved the existence of strong solutions to the obstacle problem (1.2) in [FGN12] we were able to carry out the proofs using less restrictive assumptions than in the present paper. The main difference is that, unlike in [FGN12], we here assume that the vector fields {X 1 , ..., X q } are generators of the first layer of a stratified, homogeneous group. In addition we assume that {X 1 , ..., X q } are left invariant and homogeneous of degree one.…”

“…In [FGN12] we did not have to restrict ourselves to this case and the main tool for carrying out the proofs was the lifting-approximation technique of Rotschild and Stein [RS76]. The main difficulties to overcome in the present paper, if one should try to relax these assumptions, are that of polynomial approximations and scaling, see Section 2.2 and Section 2.3 respectively.…”

“…When we began our study in [FGN12] certain estimates for the operator in (7.1) were missing, in particular Schauder type estimates, interior S p -estimates and embedding theorems corresponding to Theorems 1.2-1.4 in [FGN12]. However, for operators (1.1) at least the necessary Schauder type estimates were established and so we chose to work with (1.1) instead of (7.1).…”

“…The purpose of the paper is to advance the mathematical theory for the obstacle problem in (1.2) and in particular to continue the study of the obstacle problem initiated in [FGN12] where a number of important steps were taken towards developing a rigorous existence theory for the problem in (1.2). The main result in [FGN12] is the existence and uniqueness of a strong solution to the problem in (1.2) in certain bounded cylindrical domains D. Recall that we say that u ∈ S 1 X,loc (D) ∩ C(D) is a strong solution to problem (1.2) if the differential inequality is satisfied a.e. in D and the boundary data is attained at any point of ∂ p D. Here C(D) is the space of functions, continuous on D, and the Sobolev-Stein space S 1 X,loc (D) is defined in Section 2.…”

“…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 [FGN12] we in this paper assume, in addition, that there exists a homogeneous Lie group G = (R n , •, δ λ ) such that X1, . .…”