Orlicz-Sobolev embeddings, extensions and Orlicz-Poincaré inequalities

We provide necessary conditions on the regularity of domains for the optimal embeddings of first order (and higher order) Orlicz–Sobolev spaces into Orlicz spaces in the sense of [5], [6] (and [9]). We show that if A(t)≤C0tp near infinity for some p≥1 and W1,A(Ω)↪LAn(Ω), then there exists a constant C such that for every x∈Ω‾ and 0n, reduces to the measure density condition. A related condition that implies the measure density condition also in the critical case, is given by means of Orlicz–Poincaré inequalities.