发表机构
Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学促进学会数理研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究对数容量与扭转刚度乘积的泛函，利用几何不等式证明圆盘是平面凸集中的唯一极大值点，并建立形状可微性定理以证明非临界情形下极大值点的正则性。
AI 中文摘要
我们研究了一族涉及对数容量与扭转刚度乘积的泛函。利用若干几何不等式，我们确定了在特定参数值下，圆盘是平面凸集中唯一的极大值点。最后，我们证明了非临界情形下极大值点的正则性。这是通过在平面凸体类中建立对数容量的形状可微性定理来实现的。
英文摘要
We study a family of functionals involving a product of logarithmic capacity and torsional rigidity. Using several geometric inequalities, we identify the discs as the only maximizers among planar convex sets for certain values of the parameters. Finally, we prove regularity of maximizers in the non-critical case. This is done by establishing a shape-differentiability theorem for the logarithmic capacity in the class of planar convex bodies.
Comments20 pages, comments welcome!
