Nonlocal approximations to anisotropic Sobolev norms

TL;DR

This paper introduces nonlocal characterizations for variable exponent Sobolev spaces, relevant in nonlinear elasticity, electrorheological fluids, and image processing, especially where the exponent reaches 1.

Contribution

It provides new nonlocal approximation methods for anisotropic Sobolev norms with variable exponents, extending existing theories to critical regions.

Findings

Nonlocal characterizations for variable exponent Sobolev spaces.

Applicable to nonlinear elasticity and electrorheological fluids.

Addresses regions where the exponent reaches 1.

Abstract

We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity, in the theory of electrorheological fluids as well as in image processing for the regions where the variable exponent $p(x)$ reaches the value $1$.