Vectorial nonlinear potential theory

Abstract

We settle the longstanding problem of establishing pointwise potential estimates for vectorial solutions $u:\Omega \to {\mathbb{R}}^N$ to the non-homogeneous $p$-Laplacean system

where $\mu$ is a ${\mathbb{R}}^N$-valued measure with finite total mass. In particular, for solutions $u\in W^{1,1}({\mathbb{R}}^n)$, the global estimates via Riesz and Wolff potentials

and

respectively, hold at every point $x_0$ such that the corresponding potentials are finite. The estimates allow to give sharp descriptions of fine properties of solutions which are the exact analog of the ones in classical linear potential theory. For instance, sharp characterisation of Lebesgue points of $u$ and $Du$ and optimal regularity criteria for solutions are provided exclusively in terms of potentials.