Gradient regularity via rearrangements for p-Laplacian type elliptic boundary value problems

A sharp estimate for the distribution function of the gradient of solutions to a class of nonlinear Dirichlet and Neumann elliptic boundary value problems is established under weak regularity assumptions on the domain. As a consequence, the problem of gradient bounds in norms depending on global integrability properties is reduced to one-dimensional Hardy-type inequalities. Applications to gradient estimates in Lebesgue, Lorentz, Zygmund, and Orlicz spaces are presented. The research that led to the present paper was partially supported by a grant of the group GNAMPA of INdAM.