Variable Exponent Sobolev Spaces and Regularity of Domains

Przemysław Górka,Daniel J Pons,Nijjwal Karak

doi:10.1007/s12220-020-00499-8

Abstract

We study the embeddings of variable exponent Sobolev and Hölder function spaces over Euclidean domains, providing necessary and/or sufficient conditions on the regularity of the exponent and/or the domain in various contexts. Concerning the exponent, the relevant condition is log-Hölder continuity; concerning the domain, the relevant condition is the measure density condition.

Highlights

1 Introduction In Euclidean domains, variable exponent Lebesgue–Sobolev and Hölder spaces have been intensively studied during the last years
We consider inclusions between Lebesgue, Sobolev, and Hölder spaces with variable exponent on Euclidean domains, and obtain sufficient and/or necessary conditions on the regularity of the exponent and/or on the domain
To have a point of reference, we overview the problem concerning the density of smooth functions in variable and classical Sobolev spaces; in the variable exponent case, at least when the domain is regular enough, the regularity of the exponent is fundamental: the log-Hölder regularity turns out to be a sufficient condition

Summary

Introduction

In Euclidean domains, variable exponent Lebesgue–Sobolev and Hölder spaces have been intensively studied during the last years (see the books [3,5] for a gentle introduction, and [6] for an overview on the history of the subject) These spaces of functions provide a useful tool for the description of non-linear phenomena in elastic mechanics [21], fluid mechanics [18] and image restoration [16], for example. We describe the (nowadays) canonical notions of regularity for domains and exponents in detail, and quote the known results for embeddings between Sobolev and/or Hölder spaces using these notions After this has been done, we state our main results, that could be seen as a mixture of sufficient conditions on the regularity of the exponent, and necessary conditions on the regularity of the domain.

Preliminaries

Conclusion:

Main Results

Conclusion

Consequences and Questions