k-Uniform Rotundity of Orlicz-Sobolev Spaces

It is well known that the theory of Orlicz-Sobolev Spaces is important in dealing with non-linear Partial Differential Equations. This paper gives a necessary and sufficient condition for the uniform rotundity of Orlicz-Sobolev Spaces. Criteria for Orlicz-Sobolev Spaces to be reflexive and to have no subspaces isomorphic to co, l∞, l1 are given. Definition 1. Let A(u )= � |u| 0 p(t)dt ,w herep(t )s atisf ies: (1) p(t) is right-continuous and non-decreasing, (2) p(t) > 0, ( t> 0) (3) p(0) = 0, limt→∞ p(t )= ∞. Then A(u) is called an N-function and p(t) is called the right derivative of A(u). Definition 2. Let A(u) be an N-function, p(t )b e the right derivative of A(u). We define q(v )=s up{u ≥ 0: p(u) ≤ v} =i nf{u ≥ 0: p(u) >v }.