Partial regularity and potentials

Tuomo Kuusi, Giuseppe Mingione

Research output: Contribution to journalArticleScientificpeer-review

5 Citations (Scopus)
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Abstract

We connect classical partial regularity theory for elliptic systems to Nonlinear Potential Theory of possibly degenerate equations. More precisely, we find a potential theoretic version of the classical ε-regularity criteria leading to regularity of solutions of elliptic systems. For non-homogenous systems of the type −div a(Du) = f, the new ε-regularity criteria involve both the classical excess functional of Du and optimal Riesz type and Wol potentials of the right hand side f. When applied to the homogenous case −div a(Du) = 0 such criteria recover the classical ones in partial regularity. As a corollary, we find that the classical and sharp regularity results for solutions to scalar equations in terms of function spaces for f extend verbatim to general systems in the framework of partial regularity, i.e. optimal regularity of solutions outside a negligible, closed singular set. Finally, the new ε-regularity criteria still allow to provide estimates on the Hausdor dimension of the singular sets.

Original languageEnglish
Pages (from-to)309-363
Number of pages55
JournalJournal de l'Ecole Polytechnique - Mathematiques
Volume3
DOIs
Publication statusPublished - 2016
MoE publication typeA1 Journal article-refereed

Keywords

  • Elliptic system
  • Nonlinear potential theory
  • Partial regularity
  • Ε-regularity

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