A Pólya–Szegö principle for general fractional Orlicz–Sobolev spaces

Abstract

In this article, we prove modular and norm Pólya–Szegö inequalities in general fractional Orlicz–Sobolev spaces by using the polarization technique. We introduce a general framework which includes the different definitions of these spaces in the literature, and we establish some of its basic properties such as the density of smooth functions. As a corollary, we prove a Rayleigh–Faber–Krahn type inequality for Dirichlet eigenvalues under nonlocal nonstandard growth operators.

Keywords:

• Orlicz–Sobolev spaces
• Pólya–Szegö type principle
• rearrangement inequalities

AMS SUBJECT CLASSIFICATIONS:

• 46E30
• 35R11
• 45G05

Acknowledgments

All of the authors are members of CONICET.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This paper is partially supported by the Secretaria de Ciencia y Tecnica, Universidad de Buenos Aires (UBACyT) [grants numbers 20020130100283BA and 20020160100002BA], Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) [grant numbers PIP 11220150100032CO and 11220130100006CO], and by ANPCyT [grant numbers PICT 2012-0153 and 2014-1771].