标题： 关于在$BV(\mathbb B^n)$中具有非标准归一化的庞加莱迹线不等式 显示英文标题

标题： Poincaré trace inequalities in $BV(\mathbb B^n)$ with nonstandard normalization

摘要： 单位球 ${\mathbb B}^n$ 中有界变差函数的庞加莱迹线不等式中展示了极值函数，该单位球位于 $n$-维欧几里得空间 ${\mathbb R}^n$ 中。 试验函数要么满足平均值为零的条件，要么在整个 ${\mathbb B}^n$ 中满足中位数为零的条件，而不是像通常那样仅在 $\partial {\mathbb B}^n$ 上满足这些条件。 根据施加的约束条件，所讨论的极值函数具有不同的形式。 特别是，在后一种约束下，会出现形状异常的极值函数。 我们方法中的关键步骤是通过 $\Omega$ 的子集的等周不等式来刻画任意允许域 $\Omega \subset {\mathbb R}^n$ 中相关迹线不等式的最佳常数。 显示英文摘要

摘要： Extremal functions are exhibited in Poincar\'e trace inequalities for functions of bounded variation in the unit ball ${\mathbb B}^n$ of the $n$-dimensional Euclidean space ${\mathbb R}^n$. Trial functions are subject to either a vanishing mean value condition, or a vanishing median condition in the whole of ${\mathbb B}^n$, instead of just on $\partial {\mathbb B}^n$, as customary. The extremals in question take a different form, depending on the constraint imposed. In particular, under the latter constraint, unusually shaped extremal functions appear. A key step in our approach is a characterization of the sharp constant in the relevant trace inequalities in any admissible domain $\Omega \subset {\mathbb R}^n$, in terms of an isoperimetric inequality for subsets of $\Omega$.