Brezis Open Problem 2.2

Let $B_1\subset\mathbb{R}^2$ be the unit disk, let $\varepsilon>0$ , and define the Ginzburg--Landau energy $E_\varepsilon(u)=\int_{B_1}\left(\frac12|\nabla u|^2+\frac{1}{4\varepsilon^2}(1-|u|^2)^2\right)\,dx$ for $u\in H^1(B_1;\mathbb{R}^2)$ . Let $U_\varepsilon(x)=f_\varepsilon(|x|)\,x/|x|$ be the degree-one radial solution of the planar Ginzburg--Landau equation with boundary trace $U_\varepsilon|_{\partial B_1}(x)=x$ . Is $U_\varepsilon$ a global minimizer of $E_\varepsilon$ among all $u\in H^1(B_1;\mathbb{R}^2)$ satisfying $u|_{\partial B_1}(x)=x$ ?

Sources & referencesView supporting material

Primary source

arXiv

Additional references

On Brezis' open problem 2.2 — arXiv — Hong-Ge Chen, Yong Liu, Juncheng Wei, Wen Yang

calculus-of-variations partial-differential-equations

2 days ago

Progress summary

Refreshed 2 days ago

Claimed solved

An unrefereed manuscript claims to settle Brezis’s question for every parameter value, but the proof has not yet been independently verified.

Brezis asked whether the degree- $1$ radial Ginzburg–Landau solution is the global minimizer in the unit disk with boundary data $u(x)=x$ . The question was first numbered Open Problem $6$ and later Open Problem $2.2$ .

Known results

Strict convexity settles $\varepsilon\geq\lambda_1^{-1/2}$ .
Pacard–Rivière proved that, for sufficiently small $\varepsilon$ , the radial solution is the only critical point.
These results left intermediate values of $\varepsilon$ unresolved.

August 2026 claimed solution

Hong-Ge Chen, Yong Liu, Juncheng Wei, and Wen Yang claim that, for every $\varepsilon>0$ , $U_\varepsilon(r,\theta)=f_\varepsilon(r)e^{i\theta}$ globally minimizes the energy among maps on $B_1(0)$ with boundary value $u(x)=x$ , uniquely up to almost-everywhere equality. Their argument compares with the whole-plane vortex and decomposes the difference into nonnegative Fourier-mode contributions. No referee report, independent verification, counterexample, or withdrawal was found.

Current status (as of August 2026): The all-parameter minimization is claimed in an unrefereed version- $1$ preprint, while the claim remains unverified; the result is limited to the disk and the stated boundary condition.

Sources

arxiv.org

No solutions have been posted yet.