Calculus of Variations and Geometric Measure Theory
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G. Mingione

Gradient estimates below the duality exponent

created by mingione on 29 Sep 2012
modified on 30 Sep 2012

[BibTeX]

Published Paper

Inserted: 29 sep 2012
Last Updated: 30 sep 2012

Journal: Math. Ann.
Volume: 346
Pages: 571-627
Year: 2010

Abstract:

We show sharp local a priori estimates and regularity results for possibly degenerate non-linear elliptic problems, with data not lying in the natural dual space. We provide a precise non-linear potential theoretic analog of classical potential theory results due to Adams (Duke Math J 42:765–778, 1975) and Adams and Lewis (Studia Math 74:169–182, 1982), concerning Morrey spaces imbedding-regularity properties. For this we introduce a technique allowing for a “non-local representation” of solutions via Riesz potentials, in turn yielding optimal local estimates simultaneously in both rearrangement and non-rearrangement invariant function spaces. In fact we also derive sharp estimates in Lorentz spaces, covering borderline cases which remained open for some while.

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