On the extension problem for semiconcave functions with fractional modulus

TL;DR

This paper proves that semiconcave functions with fractional modulus defined on domains with $C^{1,1}$ boundary can be extended beyond the boundary while preserving their fractional semiconcavity, enabling new approximation and singularity analysis methods.

Contribution

It establishes the extension property for fractional semiconcave functions at boundary points, a novel result with applications in approximation and singularity propagation.

Findings

Semiconcave functions with fractional modulus can be extended beyond boundary points.

The extension preserves the fractional semiconcavity modulus.

Applications include improved approximation and analysis of singularities at boundaries.

Abstract

Consider a locally Lipschitz function $u$ on the closure of a possibly unbounded open subset $\Omega$ of $\mathbb{R}^n$ with $C^{1,1}$ boundary. Suppose $u$ is semiconcave on $\overline \Omega$ with a fractional semiconcavity modulus. Is it possible to extend $u$ in a neighborhood of any boundary point retaining the same semiconcavity modulus? We show that this is indeed the case and we give two applications of this extension property. First, we derive an approximation result for semiconcave functions on closed domains. Then, we use the above extension property to study the propagation of singularities of semiconcave functions at boundary points.