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 formF(u, Omega) = integral(Omega)(F(x, Du(x)) + f(x)) . u(x))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 bar right arrow 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.