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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Survey on gradient estimates for nonlinear elliptic equations in various function spaces

Authors: S.-S. Byun, D. K. Palagachev and L. G. Softova
Original publication: Algebra i Analiz, tom 31 (2019), nomer 3.
Journal: St. Petersburg Math. J. 31 (2020), 401-419
MSC (2010): Primary 35J60, 35R05; Secondary 35B65, 46E30, 46E35
Published electronically: April 30, 2020
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Abstract: Very general nonvariational elliptic equations of $ p$-Laplacian type are treated. An optimal Calderón-Zygmund theory is developed for such a nonlinear elliptic equation in divergence form in the setting of various function spaces including Lebesgue spaces, Orlicz spaces, weighted Orlicz spaces, and variable exponent Lebesgue spaces. The addressed arguments also apply to Morrey spaces, Lorentz spaces and generalized Orlicz spaces.

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Additional Information

S.-S. Byun
Affiliation: Department of Mathematics and Research Institute of Mathematics, Seoul National University, Seoul 151-747, Korea

D. K. Palagachev
Affiliation: Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, 70125 Bari, Italy

L. G. Softova
Affiliation: Department of Mathematics, University of Salerno, 84084 Fisciano, Italy

Keywords: Gradient estimate, nonlinear elliptic equation, $L^p$ space, weighted Lebesgue space, Orlicz space, BMO, Muckenhoupt weight, Reifenberg flat domain
Received by editor(s): October 8, 2018
Published electronically: April 30, 2020
Additional Notes: S.-S. Byun was supported by NRF-2015R1A4A1041675. D. K. Palagachev and L. G. Softova are members of INdAM/GNAMPA
Dedicated: Dedicated to the memory of S. G. Mikhlin
Article copyright: © Copyright 2020 American Mathematical Society