The obstacle problem for subelliptic non-divergence form operators on homogeneous groups

TL;DR

This paper proves the existence and uniqueness of strong solutions to the obstacle problem for subelliptic operators in non-divergence form on homogeneous groups, establishing regularity and embedding results under Hörmander’s condition.

Contribution

It introduces a framework for solving the obstacle problem for subelliptic operators on homogeneous groups, including new embedding theorems and regularity results.

Findings

Existence and uniqueness of strong solutions

Solutions are H"older continuous

Established embedding theorem for the operator class

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 = \{X_0, X_1, ...,X_q\}, q \le n, satisfying H\"ormanders finite rank condition. In this setting, X_0 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 {X_0, X_1, ...,X_q} generates a homogeneous Lie group. Furthermore, we prove that any strong solution belongs to a suitable class of H\"older continuous functions.