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 $\varepsilon_*:=\lambda_1^{-1/2}$. We prove that there exists $\delta>0$ such that the radial solution is the unique weak solution for every $\varepsilon\in(\varepsilon_*-\delta,\infty)$. More generally, we establish the analogous uniqueness result for the Ginzburg--Landau system on bounded connected $C^{1,1}$ domains in $\mathbb R^N$, $2\leq N\leq4$, with nontrivial boundary data. In particular, uniqueness persists slightly below the convexity threshold $\varepsilon_*$, where the strict convexity argument is no longer available. The proof combines strict convexity for $\varepsilon\geq\varepsilon_*$ with a compactness argument, nondegeneracy of the solution at $\varepsilon=\varepsilon_*$ and the implicit function theorem.