Gradient regularity for solutions to quasilinear elliptic equations in the plane

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