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Gradient estimates below the duality exponent. (English) Zbl 1193.35077
Author’s abstract: We show sharp local a priori estimates and regularity results for possibly degenerate nonlinear 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 D. R. Adams [Duke Math. J. 42, 765–778 (1975; Zbl 0336.46038)] and D. R. Adams and J. L. Lewis [Stud. Math. 74, 169–182 (1982; Zbl 0527.46022)], 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.

MSC:
35J70 Degenerate elliptic equations
35J60 Nonlinear elliptic equations
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
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