Generalized Orlicz spaces and related PDE

We prove the boundedness of the maximal operator in generalized Orlicz spaces defined on subsets of Rn. 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 esta...

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Bibliographic Details
Published in:Nonlinear analysis 2016-09, Vol.143, p.155-173
Main Authors: Harjulehto, Petteri, Hästö, Peter, Klén, Riku
Format: Article
Language:English
Subjects:
Density
Differential equations
Dirichlet energy integral
Existence of solution
Mathematical analysis
Musielak–Orlicz spaces
Nonlinearity
Nonstandard growth
Operators
Orlicz space
Poincaré inequality
Sobolev space
Variable exponent
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Summary:We prove the boundedness of the maximal operator in generalized Orlicz spaces defined on subsets of Rn. 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.
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2016.05.002