Global regularity and free boundary geometry in the planar Chon\'e-Rochet model

TL;DR

This paper investigates the regularity and free boundary geometry of minimizers in the planar Choné-Rochet model, establishing optimal regularity results and analyzing the structure of the free boundary in convex domains.

Contribution

It proves global $C^1$ and $C^{1,1}$ regularity results for minimizers, constructs examples showing optimality of regularity, and demonstrates the $C^1$ regularity of the free boundary.

Findings

Global $C^1$ regularity on arbitrary convex domains

Global $C^{1,1}$ regularity on strictly convex domains

Free boundary is locally a $C^1$ embedded curve

Abstract

In this paper, we study minimizers of the Chon\'e--Rochet variational problem in dimension two. We first establish global $C^1$ regularity on arbitrary bounded convex domains, and then prove global $C^{1,1}$ regularity on bounded strictly convex domains or, more generally, whenever the zero set of $u$ has positive measure. Next, we construct smooth bounded convex domains with a flat boundary segment for which no prescribed modulus of continuity controls the gradient; this shows that, without additional geometric assumptions, global $C^1$ regularity is optimal. Finally, we prove that the tamed free boundary (that is, the interface between the strictly convex and non-strictly convex regions of the solution) is locally a $C^1$ embedded curve, significantly strengthening previously known regularity results.