Abstract
In this paper, we study the extreme points and rotundity of Orlicz-Sobolev spaces. Analyzing and combining the properties of both Orlicz spaces and Sobolev spaces, we get the sufficient and necessary criteria for Orlicz-Sobolev spaces equipped with a modular norm to be uniformly rotund in every direction.
Highlights
Sobolev spaces are valuable mathematical models which were formed in the th century
We discuss the extreme points and give the sufficient and necessary criteria for Orlicz-Sobolev spaces equipped with a modular norm to be uniformly rotund in every direction
We introduce the Orlicz-Sobolev space, which is an expansion of the Orlicz space
Summary
Sobolev spaces are valuable mathematical models which were formed in the th century. Chen and Hu discussed the extreme points and rotundity of Orlicz-Sobolev spaces with maximum norm and Luxemburg norm (see [ , ]). We can study the Orlicz-Sobolev spaces with modular norm by using the methods of Orlicz spaces. We discuss the extreme points and give the sufficient and necessary criteria for Orlicz-Sobolev spaces equipped with a modular norm to be uniformly rotund in every direction. ([ ]) Suppose M is strictly convex, for any D > , ε > , there exists δ > such that, for any u, v, satisfying |u| ≤ D, |v| ≤ D, |u – v| ≥ ε, we have u+v M. The Orlicz-Sobolev space is defined as follows:.
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