Gradient regularity for solutions to quasilinear elliptic equations in the plane

We investigate the Dirichlet problem{−diva(x,∇v)=finΩv=0on∂Ω for a quasilinear elliptic equation in a planar domain Ω, when f   belongs to the Zygmund space L(logL)12(loglogL)ϵ(Ω), 0<ϵ<10<ϵ<1. We prove that the gradient of the variational solution v∈W01,2(Ω) belongs to the space L2(loglogL)2ϵ(Ω;R2). A main tool is a result on the regularity of the gradient of the solution φ to the Dirichlet problem{diva(x,∇φ)=divχ̲inΩφ∈W01,1(Ω) where χ̲∈L2(loglogL)−β(Ω;R2), β>0β>0. Namely, if the mapping a:Ω×R2→R2a:Ω×R2→R2 satisfies the Leray–Lions type conditions, then we prove the estimates‖∇φ‖L2(loglogL)−β(Ω;R2)⩽C(β)‖χ̲‖L2(loglogL)−β(Ω;R2) by applying a method recently suggested by L. Greco et al., which is based on the uniform estimates‖∇φ‖L2−σ(Ω;R2)⩽C‖χ̲‖L2−σ(Ω;R2) available for |σ|⩽σ0|σ|⩽σ0 provided that χ̲∈L2−σ(Ω;R2).