标题： 关于广义多面体网格上分片$W^{1,p}$函数的 Sobolev-Poincaré 不等式 显示英文标题

标题： Sobolev-Poincaré inequalities for piecewise $W^{1,p}$ functions over general polytopic meshes

摘要： 我们建立了Sobolev-Poincaré不等式，适用于任意维度中由相当一般的多边形网格（因此也包括形状规则的单纯形和笛卡尔网格）序列上的分片$W^{1,p}$函数；其中包括分片$W^{1,p}$函数的标准Poincaré不等式的特殊情况，并且对于非协调有限元离散化的非线性问题分析可能有用。 推导这些不等式的关键工具是新的带混合边界条件的Sobolev-迹不等式和Babuška-Aziz不等式。 我们提供的估计不含常数项，即完全显式地依赖于区域的几何性质以及底层多边形网格序列的特性。 显示英文摘要

摘要： We establish Sobolev-Poincar\'e inequalities for piecewise $W^{1,p}$ functions over sequences of fairly general polytopic (thence also shape-regular simplicial and Cartesian) meshes in any dimension; amongst others, they cover the case of standard Poincar\'e inequalities for piecewise $W^{1,p}$ functions and can be useful in the analysis of nonconforming finite element discretizations of nonlinear problems. Crucial tools in their derivation are novel Sobolev-trace inequalities and Babu\v ska-Aziz inequalities with mixed boundary conditions. We provide estimates that are constant free, i.e., that are fully explicit with respect to the geometric properties of the domain and the underlying sequence of polytopic meshes.