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 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.
Botti et al. (Fri,) studied this question.