An end point Orlicz type estimate for nonlinear elliptic equations. (December 2018)

• Record Type: Journal Article
• Authors: Jang, Yunsoo; Kim, Youchan
• Is Part Of: Nonlinear analysis. Volume 177(2018)Part B
• Page Start: 572
• Page End: 585
• Publication Date: 2018-12
• Subjects: primary 35J60 35B65 -- secondary 35R05; Calderón–Zygmund type estimates -- Nonlinear elliptic equations -- Measurable nonlinearities; 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.2018.03.014
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study nonlinear elliptic equations of p -Laplacian type in divergence form to establish a natural Calderón–Zygmund type theory of an Orlicz space type, where the Lebesgue space is the special case with Φ ( t ) = t q p : Φ ( | F | p ) ∈ L 1 ( Q 2 R ) ⟹ Φ ( | D u | p ) ∈ L 1 ( Q R ) . In the previous results, the estimates obtained were strictly above the natural exponents, and the function such as Φ ( t ) = t log ( 1 + t ) was ruled out for the candidate of Φ . But with our approach, Φ can be selected up to the end point case of the estimates, and the functions Φ ( t ) = t and Φ ( t ) = t log ( 1 + t ) can be additionally selected for Φ .