Sobolev–Poincaré Inequalities for Piecewise 𝐖1,𝑝 Functions over General Polytopic Meshes

We establish Sobolev–Poincaré inequalities for piecewise W1,𝑝 functions over families of fairly general polytopic (thence also shape-regular simplicial and Cartesian) meshes in any dimension; among others, they cover the case of standard Poincaré inequalities for piecewise W1,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 W1,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.