We deal with Orlicz-Sobolev embeddings in open subsets of R. A necessary and sufficient condition is established for the existence of an optimal, i.e. largest possible, Orlicz-Sobolev space continuously embedded into a given Orlicz space. Moreover, the optimal Orlicz-Sobolev space is exhibited whenever it exists. Parallel questions are addressed for Orlicz-Sobolev embeddings into Orlicz spaces ...

Recently, Gardner, Hug and Weil developed an Orlicz-Brunn1 Minkowski theory. Following this, in the paper we further consider the 2 Orlicz-Brunn-Minkowski theory. The fundamental notions of mixed quer3 massintegrals, mixed p-quermassintegrals and inequalities are extended to 4 an Orlicz setting. Inequalities of Orlicz Minkowski and Brunn-Minkowski 5 type for Orlicz mixed quermassintegrals are o...

converge a.e. for all f in L log log(L) but fail to have a finite limit for an f ∈ L. In fact, we show that for each Orlicz space properly contained in L, 1 ≤ q < ∞, there is a sequence along which the ergodic averages converge for functions in the Orlicz space, but diverge for all f ∈ L . This extends the work of K. Reinhold, who, building on the work of A. Bellow, constructed a sequence for w...

In this article, we consider three types of solutions in Orlicz spaces for the quasilinear elliptic problem − div(a(|∇u|)∇u) = 0. By applying a comparison principle, we establish the relationships between viscosity supersolutions, weak supersolutions, and superharmonic functions.