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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.23682 |
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| _version_ | 1866917437285335040 |
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| author | Chen, Shibing Li, Yuanyuan Wang, Xianduo |
| author_facet | Chen, Shibing Li, Yuanyuan Wang, Xianduo |
| contents | We study the free-boundary equation \[
Δu=χ_{\{|\nabla u|>0\}} \] near the origin. We prove that, at a singular point of \(\partial\{|\nabla u|>0\}\), the quadratic blow-up is unique. As noted in \cite[Notes to Chapter 7]{PSU2012}, little is known about the singular set for this problem. The usual Weiss--Monneau monotonicity argument does not seem to apply directly, because the inactive set is determined by the vanishing of \(\nabla u\), rather than by a sign condition on \(u\). The proof follows the quadratic part of the rescalings. Projecting onto the trace-free quadratic harmonics yields a finite-dimensional differential equation for the quadratic coefficient. Together with a Lyapunov identity and estimates on dyadic annuli, this implies convergence of the quadratic coefficient, and hence uniqueness of the blow-up. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23682 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uniqueness of Blow-ups for the Superconductivity Free Boundary Problem Chen, Shibing Li, Yuanyuan Wang, Xianduo Analysis of PDEs We study the free-boundary equation \[ Δu=χ_{\{|\nabla u|>0\}} \] near the origin. We prove that, at a singular point of \(\partial\{|\nabla u|>0\}\), the quadratic blow-up is unique. As noted in \cite[Notes to Chapter 7]{PSU2012}, little is known about the singular set for this problem. The usual Weiss--Monneau monotonicity argument does not seem to apply directly, because the inactive set is determined by the vanishing of \(\nabla u\), rather than by a sign condition on \(u\). The proof follows the quadratic part of the rescalings. Projecting onto the trace-free quadratic harmonics yields a finite-dimensional differential equation for the quadratic coefficient. Together with a Lyapunov identity and estimates on dyadic annuli, this implies convergence of the quadratic coefficient, and hence uniqueness of the blow-up. |
| title | Uniqueness of Blow-ups for the Superconductivity Free Boundary Problem |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.23682 |