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PreprintCC BY 4.0arXiv

A partial answer to Brezis' Open Problem 2.1

  • Chao Ji
  • Kai Sheng
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Abstract

In this paper, we give a partial answer to Brezis' Open Problem~2.1, which concerns the uniqueness of solutions to the Ginzburg--Landau equation in the unit disc with the degree-one boundary condition. Let λ1 be the first Dirichlet eigenvalue of −Δ in the unit disc and set ε*:=λ1−1/2. We prove that there exists δ>0 such that the radial solution is the unique weak solution for every ε∈(ε*−δ,∞). More generally, we establish the analogous uniqueness result for the Ginzburg--Landau system on bounded connected C1,1 domains in ℝN, 2≤ N≤4, with nontrivial boundary data. In particular, uniqueness persists slightly below the convexity threshold ε*, where the strict convexity argument is no longer available. The proof combines strict convexity for ε≥ε* with a compactness argument, nondegeneracy of the solution at ε=ε* and the implicit function theorem.

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Cite

BibTeXRIS
BibTeX
@misc{ji2026partial,
  title = {{A partial answer to Brezis' Open Problem 2.1}},
  author = {Chao Ji and Kai Sheng},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.21896},
  eprint = {2608.21896},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.21896},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A partial answer to Brezis' Open Problem 2.1
AU  - Chao Ji
AU  - Kai Sheng
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2608.21896
UR  - https://arxiv.org/abs/2608.21896
AB  - In this paper, we give a partial answer to Brezis' Open Problem~2.1, which concerns the uniqueness of solutions to the Ginzburg--Landau equation in the unit disc with the degree-one boundary condition. Let λ_1 be the first Dirichlet eigenvalue of −Δ in the unit disc and set ε_*:=λ_1^−1/2. We prove that there exists δ>0 such that the radial solution is the unique weak solution for every ε∈(ε_*−δ,∞). More generally, we establish the analogous uniqueness result for the Ginzburg--Landau system on bounded connected C^1,1 domains in ℝ^N, 2≤ N≤4, with nontrivial boundary data. In particular, uniqueness persists slightly below the convexity threshold ε_*, where the strict convexity argument is no longer available. The proof combines strict convexity for ε≥ε_* with a compactness argument, nondegeneracy of the solution at ε=ε_* and the implicit function theorem.
ER  -