Very degenerate elliptic equations under almost critical Sobolev regularity

Abstract

Abstract We prove the local Lipschitz continuity and the higher differentiability of local minimizers of functionals of the form 𝔽 ⁢ ( u , Ω ) = ∫ Ω ( F ⁢ ( x , D ⁢ u ⁢ ( x ) ) + f ⁢ ( x ) ⋅ u ⁢ ( x ) ) ⁢ d ⁢ x \mathbb{F}(u,\Omega)=\int_{\Omega}(F(x,Du(x))+f(x)\cdot u(x))\mathop{}\!dx with non-autonomous integrand F ⁢ ( x , ξ ) {F(x,\xi)} which is degenerate convex with respect to the gradient variable. The main novelty here is that the results are obtained assuming that the partial map x ↦ D ξ ⁢ F ⁢ ( x , ξ ) {x\mapsto D_{\xi}F(x,\xi)} has weak derivative in the almost critical Zygmund class L n ⁢ log α ⁡ L {L^{n}\log^{\alpha}L} and the datum f is assumed to belong to the same Zygmund class.