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Gradient estimates below the duality exponent

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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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  1. Dipartimento di Matematica, Università di Parma, Viale G.P. Usberti 53/a, Campus, 43100, Parma, Italy

    Giuseppe Mingione

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Correspondence to Giuseppe Mingione.

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This paper is dedicated to Carlo Sbordone on his 60th birthday.

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Mingione, G. Gradient estimates below the duality exponent. Math. Ann. 346, 571–627 (2010). https://doi.org/10.1007/s00208-009-0411-z

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