Distance functions with dense singular sets

TL;DR

This paper characterizes when the singular set of a distance function from a smooth hypersurface is dense, using an inner ball condition, and explores the existence of viscosity solutions with non-dense singular sets.

Contribution

It provides a characterization of the denseness of the singular set of distance functions and investigates viscosity solutions with non-dense singular sets.

Findings

Denseness of the singular set is characterized by an inner ball condition.

Conditions under which viscosity solutions have non-dense singular sets are identified.

The relationship between geometric conditions and the structure of singular sets is clarified.

Abstract

We characterize the denseness of the singular set of the distance function from a $\mathcal{C}^1$-hypersurface in terms of an inner ball condition and we address the problem of the existence of viscosity solutions of the Eikonal equation whose singular set (i.e. set of non-differentiability points) is not no-where dense.