*Published Paper*

**Inserted:** 29 sep 2012

**Journal:** Calc. Var.

**Volume:** 39

**Pages:** 379-418

**Year:** 2010

**Abstract:**

We consider degenerate elliptic equations of $p$-Laplacean type \[-\textnormal{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 $\mu$. As a corollary we identify borderline condition for the continuity of $Du$ in terms of the data: namely $\mu$ belongs to the Lorentz space $L(n,1/(p-1))$, and $\gamma(x)$ 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 $\mu$, or to the decay rate of its $L^n$-norm on small balls, are identified as well as corollaries of the main non-linear potential criterium.