Gradient continuity estimates

Abstract

We consider degenerate elliptic equations of p-Laplacean type $$-{\rm{div}}\, (\gamma(x)|Du|^{p-2}Du)=\mu\,,$$ and give a sufficient condition for the continuity of Du in terms of a natural non-linear Wolff potential of the right-hand side measure μ. As a corollary we identify borderline condition for the continuity of Du in terms of the data: namely μ belongs to the Lorentz space L(n, 1/(p − 1)), and γ(x) is a Dini continuous elliptic coefficient. This last result, together with pointwise gradient bounds via non-linear potentials, extends to the non homogeneous p-Laplacean system, thereby giving a positive answer in the vectorial case to a conjecture of Verbitsky. Continuity conditions related to the density of μ, or to the decay rate of its Ln-norm on small balls, are identified as well as corollaries of the main non-linear potential criterium.