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arXiv · 2603.21196

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

Shibing Chen·Alessio Figalli·Yi Ru-Ya Zhang
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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.

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Shibing Chen, Alessio Figalli, Yi Ru-Ya Zhang. 2026-03-22. Global regularity and free boundary geometry in the planar Chon\'e-Rochet model. https://arxiv.org/abs/2603.21196

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