Symmetry of the Ginzburg-Landau minimizer in a disc
Let \(D\) be the unit disc in \(\mathbb{R}^2\). The authors study the minimization of the Ginzburg-Landau energy \[ E(\psi)= \int_\Omega \{|\psi|^2+ J(|\psi|^2)\}, \] where \(J: [0, +\infty)\to [0, +\infty)\) satisfies the following assumptions: \hskip17mm i) \(J(0)> 0\), \(J(1)= 0\), \( J(t)\geq 0\) if \(t> 1\); \hskip17mm ii) \(J(t)\) is monotone decreasing and convex on \([0, 1]\); \hskip17mm iii) \(J\) is twice differentiable on \([0, 1]\). The main result of this paper asserts that the minimizer of \(E(\psi)\) with respect to the class \[ {\mathcal E}= \{\psi\in H^1(D); \psi(x)= x\text{ on } \partial D\} \] is stable, in the sense that the eigenvalues of the second variation operator are all nonnegative. This result gives an answer to some previous open problems raised by Jaffe-Taubes or Bethuel-Brezis-Hélein. The paper is remarkably written and the results are of the greatest interest.
scientific article; zbMATH DE number 794197