Regularity results via duality for minimizers of degenerate functionals

Abstract

We prove a regularity result for local minimizers of degenerate variational integrals, whose model arises in the study of mappings with finite distortion. The degeneracy function 𝒦(x) lies in the exponential class, i.e., exp (λ𝒦(x)) is integrable for some λ>0. The right space of the gradient of a local minimizer u turns out to be the Zygmund class L^p log ⁻¹L. Our result states that if λ is sufficiently large, then Du belongs to the Zygmund space L^p log ^αL, α≥1 and α increases with λ.