Vectorial nonlinear potential theory

  • Tuomo Kuusi

    University of Oulu, Finland
  • Giuseppe Mingione

    Università di Parma, Italy
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We settle the longstanding problem of establishing pointwise potential estimates for vectorial solutions u ⁣:ΩRNu\colon \Omega \to \mathbb R^{N} to the non-homogeneous pp-Laplacean system

div(Dup2Du)=μ\mboxin ΩRn,-\rm{div} (|Du|^{p-2}Du)=\mu \qquad \mbox{in}\ \Omega \subset \mathbb R^{n}\,,

where μ\mu is a RN\mathbb R^{N}-valued measure with finite total mass. In particular, for solutions uW1,1(Rn)u \in W^{1,1}(\mathbb R^{n}), the global estimates via Riesz and Wolff potentials

Du(x0)p1Rndμ(x)xx0n1|Du(x_0)|^{p-1} \lesssim \int_{\mathbb R^{n}}\frac{d|\mu|(x)}{|x-x_0|^{n-1}}


u(x0)W1,pμ(x0,)=0(μ(Bϱ(x0))ϱnp)1/(p1)dϱϱ|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 x0x_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 uu and DuDu 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