Vectorial nonlinear potential theory

  • Tuomo Kuusi

    University of Oulu, Finland
  • Giuseppe Mingione

    Università di Parma, Italy


We settle the longstanding problem of establishing pointwise potential estimates for vectorial solutions to the non-homogeneous -Laplacean system

where is a -valued measure with finite total mass. In particular, for solutions , the global estimates via Riesz and Wolff potentials


respectively, hold at every point 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 and and optimal regularity criteria for solutions are provided exclusively in terms of potentials.

Cite this article

Tuomo Kuusi, Giuseppe Mingione, Vectorial nonlinear potential theory. J. Eur. Math. Soc. 20 (2018), no. 4, pp. 929–1004

DOI 10.4171/JEMS/780