Partial regularity for solutions to subelliptic eikonal equations

TL;DR

This paper proves that solutions to subelliptic eikonal equations are smooth outside a negligible set, extending regularity results to systems of vector fields satisfying Hörmander's condition.

Contribution

It establishes partial regularity for solutions to subelliptic eikonal equations with Hörmander vector fields, showing smoothness outside a measure-zero set.

Findings

Solutions are smooth outside a closed measure-zero set.

The result applies to systems satisfying Hörmander's bracket generating condition.

Extends regularity theory for subelliptic PDEs.

Abstract

On a bounded domain $\Omega$ in euclidean space $\mathbb{R}^n$, we study the homogeneous Dirichlet problem for the eikonal equation associated with a system of smooth vector fields, which satisfies H\"ormander's bracket generating condition. We prove that the solution is smooth in the complement of a closed set of Lebesgue measure zero.