Very degenerate elliptic equations under almost critical Sobolev regularity

We prove the local Lipschitz continuity and the higher differentiability of local minimizers of functionals of the form\mathbb{F}(u,\Omega)=\int_{\Omega}(F(x,Du(x))+f(x)\cdot u(x))\mathop{}\!dxwith non-autonomous integrand {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\mapsto D_{\xi}F(x,\xi)} has weak derivative in the almost critical Zygmund class {L^{n}\log^{\alpha}L} and the datum f is assumed to belong to the same Zygmund class.