On optimal C^1,α estimates for p(x)-Laplace type equations

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)=divh(x)+f(x)inΩwith 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).