标题： 自旋子二维插值不等式的对称性与对称破缺——初步结果 显示英文标题

标题： Symmetry and symmetry breaking in interpolation inequalities for two-dimensional spinors -- Preliminary results

摘要： 在二维欧氏空间中，我们研究了包含加权梯度范数的Caffarelli-Kohn-Nirenberg不等式的旋量类似物。 该(SCKN)不等式等价于圆柱体上的旋量Gagliardo-Nirenberg型插值不等式，以及一个包含Aharonov-Bohm磁场的插值不等式，后者已在2020年的一篇论文中进行了分析。 我们通过围绕径向极小值器线性化相关泛函来检验最优函数的对称性。 我们证明了线性化问题的稳定性等价于$2\times2$矩阵值微分算子的正性。 我们通过结合解析论证和数值计算来研究正性问题。 具体而言，我们的结果提供了数值证据，表明对称破缺区域超出了先前已知的范围，而已知对称区域的阈值是线性稳定的。 总而言之，我们获得了对称性和对称性破缺之间相变的精确估计。 我们的结果也与最近研究的三维（SCKN）不等式存在显著差异。 显示英文摘要

摘要： On the two-dimensional Euclidean space, we study a spinorial analogue of the Caffarelli-Kohn-Nirenberg inequality involving weighted gradient norms. This (SCKN) inequality is equivalent to a spinorial Gagliardo-Nirenberg type interpolation inequality on a cylinder as well as to an interpolation inequality involving Aharonov-Bohm magnetic fields, which was analyzed in a paper of 2020. We examine the symmetry properties of optimal functions by linearizing the associated functional around radial minimizers. We prove that the stability of the linearized problem is equivalent to the positivity of a $2\times2$ matrix-valued differential operator. We study the positivity issue via a combination of analytical arguments and numerical computations. In particular, our results provide numerical evidence that the region of symmetry breaking extends beyond what was previously known, while the threshold of the known symmetry region is linearly stable. Altogether, we obtain refined estimates of the phase transition between symmetry and symmetry breaking. Our results also put in evidence striking differences with the three-dimensional (SCKN) inequality that was recently investigated.