A partial answer to Brezis' Open Problem 2.1

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 ${\lambda }_1$ be the first Dirichlet eigenvalue of $-\Delta$ in the unit disc and set ${\epsilon }_{\ast }:={\lambda }_1^{-1/2}$. We prove that there exists $\delta >0$ such that the radial solution is the unique weak solution for every $\epsilon \in ({\epsilon }_{\ast }-\delta ,\infty )$. More generally, we establish the analogous uniqueness result for the Ginzburg--Landau system on bounded connected $C^{1,1}$ domains in ${\mathbb{R}}^N$, $2\le N\le 4$, with nontrivial boundary data. In particular, uniqueness persists slightly below the convexity threshold ${\epsilon }_{\ast }$, where the strict convexity argument is no longer available. The proof combines strict convexity for $\epsilon \ge {\epsilon }_{\ast }$ with a compactness argument, nondegeneracy of the solution at $\epsilon ={\epsilon }_{\ast }$ and the implicit function theorem.

12 pages. Comments and suggestions are most welcome