Partial regularity and potentials
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A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
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Authors
Date
2016
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Language
en
Pages
55
309-363
309-363
Series
Journal de l'Ecole Polytechnique - Mathematiques, Volume 3
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.Description
Keywords
Elliptic system, Nonlinear potential theory, Partial regularity, Ε-regularity
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Citation
Kuusi, T & Mingione, G 2016, ' Partial regularity and potentials ', Journal de l'Ecole Polytechnique - Mathematiques, vol. 3, pp. 309-363 . https://doi.org/10.5802/jep.35