On optimal [formula omitted] estimates for [formula omitted]-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) = 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). [ABSTRACT FROM AUTHOR]