Sobolev-Poincar\'e inequalities for piecewise $W^{1,p}$ functions over general polytopic meshes

Abstract

We establish Sobolev-Poincar\'e inequalities for piecewise $W^{1,p}$ functions over families 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 $W^{1,p}$-stable right-inverses of the divergence satisfying mixed boundary conditions. We provide estimates with constants having an explicit dependence on the geometric properties of the domain and the underlying family of polytopic meshes.