AI 中文总结
该研究证明半径R圆盘上特定边界值的Ginzburg-Landau能量全局极小是平面Ginzburg-Landau方程一次径向解，肯定了Brezis公开问题2.2，通过傅里叶模分解与Picone型恒等式完成证明。
AI 中文摘要
我们证明，在半径为R的圆盘上，边界值为u(x)=x/|x|的Ginzburg-Landau能量的全局极小值是平面Ginzburg-Landau方程的一次径向解。这对Brezis公开问题列表中的公开问题2.2给出了肯定回答。该结果通过将圆盘内的径向解f与全平面内的一次径向解F进行比较得到：将圆盘内的竞争解乘以F/f，可在不改变边界迹的情况下利用全平面涡旋的已知极小性。两种能量的差可分解为傅里叶模，所有非零模均非负，零模则通过Picone型恒等式处理。
英文摘要
We prove that the global minimizer of the Ginzburg-Landau energy in the disk of radius $R$ with boundary value $ u(x)=\frac{x}{|x|}$ is the degree-one radial solution of the planar Ginzburg--Landau equation. This gives an affirmative answer to Open Problem~2.2 in Brezis' open-problem list. This is achieved by comparing the radial solution $f$ in the disk with the degree-one radial solution $F$ in the whole plane. Multiplying a disk competitor by $F/f$ enables us to use the known minimality of the whole-plane vortex without changing the boundary trace. The difference of the two energies can be decomposed into Fourier modes. Every nonzero mode is nonnegative, and the zero mode is then handled by a Picone type identity.
