Calculus of Variations and Geometric Measure Theory
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T. Kuusi - G. Mingione

Partial regularity and potentials

created by mingione on 27 Aug 2016

[BibTeX]

Published Paper

Inserted: 27 aug 2016

Journal: Journal de l'école polytechnique - mathématiques
Year: 2016
Doi: 10.5802/jep.35
Links: paper

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 $\varepsilon $-regularity criteria leading to regularity of solutions of elliptic systems. For non-homogenous systems of the type $-\mathrm{div}\, a(Du)=f$, the new $\varepsilon $-regularity criteria involve both the classical excess functional of $Du$ and optimal Riesz type and Wolff potentials of the right hand side $f$. When applied to the homogenous case $-\mathrm{div}\, a(Du)=0$ 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 $\varepsilon $-regularity criteria still allow to provide estimates on the Hausdorff dimension of the singular sets.

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