标题： 最优磁性矩形是一个正方形吗？ 显示英文标题

标题： Is the optimal magnetic rectangle a square?

摘要： 我们关注磁性狄利克雷拉普拉斯算子的最低特征值在矩形几何结构上的依赖关系，受齐次场的影响。我们猜想，在面积或周长约束下，正方形是全局最小值。与众所周知的无磁场类似情况相反，目前的谱问题不提供显式解。通过建立特征值的下界和上界，我们证明了弱磁场下的猜想。此外，我们将猜想的有效性与特征值的简单性以及非凸最小化问题的极小值点的对称性联系起来。 显示英文摘要

摘要： We are concerned with the dependence of the lowest eigenvalue of the magnetic Dirichlet Laplacian on the geometry of rectangles, subject to homogeneous fields. We conjecture that the square is a global minimiser both under the area or perimeter constraints. Contrary to the well-known magnetic-free analogue, the present spectral problem does not admit explicit solutions. By establishing lower and upper bound to the eigenvalue, we establish the conjecture for weak magnetic fields. Moreover, we relate the validity of the conjecture to the simplicity of the eigenvalue and symmetries of minimisers of a non-convex minimisation problem.