Distance functions with dense singular sets

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.

Keywords:

• Baire category
• convex sets
• distance function
• Eikonal equation
• singular set
• viscosity solutions

MSC-classes 2020::

• 35D40
• 35F21
• 52A20

Notes

1 This is a map $\sigma \in {\mathcal{C}}^{k,\alpha }(U,{\mathbf{R}}^n)$ such that $\sigma (U)$ is an open subset of ${\mathbf{R}}^n$ and ${\sigma }^{-1}\in {\mathcal{C}}^{k,\alpha }(\sigma (U),{\mathbf{R}}^n).$

2 Let $S\subseteq {\mathbf{R}}^n.$ Following [4, 2.1.1] we say that a function $f:S\to \mathbf{R}$ is locally semiconcave on S with linear modulus of continuity if there exists a constant C_S (semiconcavity constant for f in S) such that

$\lambda f(x)+(1-\lambda )f(y)-f((1-\lambda )x+\lambda y)\le \frac{C_S}{2}\lambda (1-\lambda )|x-y{|}^2$

for every $0\le \lambda \le 1$ and for every $x,y\in S$ such that the segment joining x with y is contained in S.

3 In fact ${\mathcal{K}}_r^n$ is a comeager of the space of all convex bodies (with non empty interior) equipped with the Haussdorf metric.