Topology of singular set of semiconcave function via Arnaud's theorem

TL;DR

This paper establishes the local path-connectedness of certain singular sets of semiconcave functions with linear modulus, using Arnaud's theorem, and provides a new proof for the time-dependent case.

Contribution

It extends the understanding of the topology of singular sets of semiconcave functions and offers a new proof for the time-dependent scenario based on Arnaud's theorem.

Findings

Proved local path-connectedness of specific singular sets.

Demonstrated the optimality of the result.

Provided a new proof for the time-dependent case.

Abstract

We proved the (local) path-connectedness of certain subset of the singular set of semiconcave functions with linear modulus in general. In some sense this result is optimal. The proof is based on a theorem by Marie-Claude Arnaud (M.-C. Arnaud, \textit{Pseudographs and the Lax-Oleinik semi-group: a geometric and dynamical interpretation}. Nonlinearity, \textbf{24}(1): 71-78, 2011.). We also gave a new proof of the theorem in time-dependent case.