标题： 等周不等式对于内平行曲线 显示英文标题

标题： Isoperimetric inequalities for inner parallel curves

摘要： 我们证明了光滑、有界且单连通区域的加权等周不等式。 更准确地说，我们证明了在周长固定的情况下，内平行曲线的惯性矩在圆盘中达到最大值。 这个不等式之前仅对凸区域已知，使我们能够将磁 Robin 拉普拉斯算子的等周不等式扩展到非凸的中心对称区域。 此外，我们将针对惯性矩（即二阶矩）的等周不等式扩展到所有小于或等于二的$p$次幂的$p$阶矩。 我们还证明，在几乎为圆形的中心对称情况下，圆盘对于所有严格小于三的$p$是严格的局部最大值，并且当$p$严格大于三时，该不等式不成立。 显示英文摘要

摘要： We prove weighted isoperimetric inequalities for smooth, bounded, and simply connected domains. More precisely, we show that the moment of inertia of inner parallel curves for domains with fixed perimeter attains its maximum for a disk. This inequality, which was previously only known for convex domains, allows us to extend an isoperimetric inequality for the magnetic Robin Laplacian to non-convex centrally symmetric domains. Furthermore, we extend our isoperimetric inequality for moments of inertia, which are second moments, to $p$-th moments for all $p$ smaller than or equal to two. We also show that the disk is a strict local maximiser in the nearly circular, centrally symmetric case for all $p$ strictly less than three, and that the inequality fails for all $p$ strictly bigger than three.