Generalized Orlicz spaces and related PDE. (September 2016)

• Record Type: Journal Article
• Authors: Harjulehto, Petteri; Hästö, Peter; Klén, Riku
• Is Part Of: Nonlinear analysis. Volume 143(2016)
• Page Start: 155
• Page End: 173
• Publication Date: 2016-09
• Subjects: 35J60 -- 46E35 -- 46E30 -- 49J40; Musielak–Orlicz spaces -- Orlicz space -- Sobolev space -- Variable exponent -- Poincaré inequality -- Dirichlet energy integral -- Existence of solution -- Nonstandard growth; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.05.002
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We prove the boundedness of the maximal operator in generalized Orlicz spaces defined on subsets of R n . The proof is based on an extension result for Φ -functions. We study generalized Sobolev–Orlicz spaces and establish density of smooth functions and the Poincaré inequality. As applications we establish the existence of solutions of the φ -Laplace equation with zero and non-zero right-hand side. Further, we systematize assumptions for Φ -functions and prove several basic tools needed for the study of differential equations of generalized Orlicz growth.