Vectorial nonlinear potential theory

Tuomo Kuusi; Giuseppe Mingione

doi:10.4171/jems/780

JournalsjemsVol. 20, No. 4pp. 929–1004

Vectorial nonlinear potential theory

Tuomo Kuusi
University of Oulu, Finland
Giuseppe Mingione
Università di Parma, Italy

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Abstract

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

-\rm{div} (|Du|^{p-2}Du)=\mu \qquad \mbox{in}\ \Omega \subset \mathbb R^{n}\,,

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

|Du(x_0)|^{p-1} \lesssim \int_{\mathbb R^{n}}\frac{d|\mu|(x)}{|x-x_0|^{n-1}}

and

|u(x_0)|\lesssim {\bf W}^{\mu}_{1, p}(x_0,\infty) = \int_0^\infty \left(\frac{|\mu|(B_\varrho(x_0))}{\varrho^{n-p}}\right)^{1/(p-1)}\, \frac{d\varrho}{\varrho}

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.

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