Partial regularity and potentials

Tuomo Kuusi; Giuseppe Mingione

doi:10.5802/jep.35

Partial regularity and potentials
[Régularité partielle et potentiels]

Tuomo Kuusi ¹ ; Giuseppe Mingione ²

¹ Aalto University, Institute of Mathematics P.O. Box 11100, FI-00076 Aalto, Finland
² Dipartimento di Matematica e Informatica, Università di Parma Parco Area delle Scienze 53/a, Campus, 43100 Parma, Italy

Journal de l’École polytechnique — Mathématiques, Tome 3 (2016), pp. 309-363.

Résumé
Abstract

Nous relions la théorie classique de la régularité partielle des systèmes elliptiques à la théorie du potentiel non linéaire d’équations éventuellement dégénérées. Plus précisément, nous donnons une version en théorie du potentiel des critères classiques d’ $ε$ -régularité de solutions des systèmes elliptiques. Pour les systèmes non homogènes du type $- div a (D u) = f$ , les nouveaux critères d’ $ε$ -régularité font intervenir à la fois la fonctionnelle classique d’excès de $D u$ et de type de Riesz optimal et les potentiels de Wolff du membre de droite $f$ . Appliqués au cas homogène $- div a (D u) = 0$ , ces critères redonnent les critères classiques en théorie de la régularité partielle. Comme corollaire, nous montrons que les résultats classiques et précisés de régularité pour les solutions d’équations scalaires en terme d’espaces de fonctions pour $f$ s’étendent mot pour mot aux systèmes généraux dans le cadre de la régularité partielle, à savoir la régularité partielle des solutions hors d’un ensemble singulier fermé négligeable. Enfin, ces nouveaux critères d’ $ε$ -régularité permettent encore d’obtenir des estimée sur la dimension de Hausdorff des ensembles singuliers.

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 $- div a (D u) = f$ , the new $ε$ -regularity criteria involve both the classical excess functional of $D u$ and optimal Riesz type and Wolff potentials of the right hand side $f$ . When applied to the homogenous case $- div a (D u) = 0$ 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 Hausdorff dimension of the singular sets.

Reçu le : 2015-11-02
Accepté le : 2016-08-03
Publié le : 2016-08-23

MR Zbl

DOI : 10.5802/jep.35

Classification : 35B65, 31C45
Keywords: Partial regularity, elliptic system, nonlinear potential theory, $\varepsilon $-regularity
Mot clés : Régularité partielle, système elliptique, théorie du potentiel non linéaire, $\varepsilon $-régularité

Affiliations des auteurs :

Tuomo Kuusi ¹ ; Giuseppe Mingione ²

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Partial regularity and potentials},
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Tuomo Kuusi; Giuseppe Mingione. Partial regularity and potentials. Journal de l’École polytechnique — Mathématiques, Tome 3 (2016), pp. 309-363. doi : 10.5802/jep.35. https://jep.centre-mersenne.org/articles/10.5802/jep.35/

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