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Partial regularity and potentials
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A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
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en
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55
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Journal de l'Ecole Polytechnique - Mathematiques, Volume 3, pp. 309-363
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.
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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