Global regularity and free boundary geometry in the planar Choné-Rochet model
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917356636209152 |
|---|---|
| author | Chen, Shibing Figalli, Alessio Zhang, Yi Ru-Ya |
| author_facet | Chen, Shibing Figalli, Alessio Zhang, Yi Ru-Ya |
| contents | In this paper, we study minimizers of the Choné--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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21196 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Global regularity and free boundary geometry in the planar Choné-Rochet model Chen, Shibing Figalli, Alessio Zhang, Yi Ru-Ya Analysis of PDEs 35R35, 49N60, 35B65, 35Q91 In this paper, we study minimizers of the Choné--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. |
| title | Global regularity and free boundary geometry in the planar Choné-Rochet model |
| topic | Analysis of PDEs 35R35, 49N60, 35B65, 35Q91 |
| url | https://arxiv.org/abs/2603.21196 |