On optimal C1, α estimates for p(x)-Laplace type equations. (November 2020)

• Record Type: Journal Article
• Authors: Ding, Mengyao; Zhang, Chao; Zhou, Shulin
• Is Part Of: Nonlinear analysis. Volume 200(2020)
• Publication Date: 2020-11
• Subjects: 35J62 -- 35J70; Elliptic p(x)-Laplacian -- Optimal Hölder exponent; 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.2020.112030
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we investigate the optimal C 1, α estimates for the elliptic p ( ⋅ ) -Laplace equation: div ( a ( x ) | ∇ u | p ( x ) − 2 ∇ u ) = div h ( x ) + f ( x ) in Ω with f ∈ L q ( ⋅ ) ( Ω ) and a, h ∈ C σ ( Ω ¯ ) . Based on a certain geometric oscillation estimate, the scaling arguments and appropriate localization technique as well as the careful analysis on the variable exponents, we exhibit how the optimal Hölder exponent of ∇ u is influenced by p ( ⋅ ), q ( ⋅ ) and σ . This work can be regarded as a natural follow up to the paper by Araújo and Zhang (in press).