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Global weighted estimates for nonlinear elliptic obstacle problems over Reifenberg domains


Authors: Sun-Sig Byun, Yumi Cho and Dian K. Palagachev
Journal: Proc. Amer. Math. Soc. 143 (2015), 2527-2541
MSC (2010): Primary 35J87; Secondary 35R05, 35J60, 35B65, 46E35
DOI: https://doi.org/10.1090/S0002-9939-2015-12458-6
Published electronically: January 21, 2015
MathSciNet review: 3326034
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Abstract: We study the obstacle problem for an elliptic equation with discontinuous nonlinearity over a nonsmooth domain, assuming that the irregular obstacle and the nonhomogeneous term belong to suitable weighted Sobolev and Lebesgue spaces, respectively, with weights taken in the Muckenhoupt classes. We establish a Calderón-Zygmund type result by proving that the gradient of the weak solution to the nonlinear obstacle problem has the same weighted integrability as both the gradient of the obstacle and the nonhomogeneous term, provided that the nonlinearity has a small BMO-semi norm with respect to the gradient, and the boundary of the domain is $ \delta $-Reifenberg flat. We also get global regularity in the settings of the Morrey and Hölder spaces for the weak solutions to the problem considered.


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Additional Information

Sun-Sig Byun
Affiliation: Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747, Korea
Email: byun@snu.ac.kr

Yumi Cho
Affiliation: Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea
Address at time of publication: School of Mathematics, Korea Institute for Advanced Study, Seoul 120-722, Korea
Email: yumicho@kias.re.kr

Dian K. Palagachev
Affiliation: Dipartimento di Meccanica, Matematica e Management (DMMM), Politecnico di Bari, 70125 Bari, Italy
Email: dian.palagachev@poliba.it

DOI: https://doi.org/10.1090/S0002-9939-2015-12458-6
Keywords: Nonlinear elliptic equation, Obstacle problem, Irregular obstacle, Calder\'on-Zygmund estimate, Muckenhoupt weight, $p$-Laplacean, BMO, Reifenberg flat domain, Morrey space
Received by editor(s): August 6, 2013
Received by editor(s) in revised form: January 15, 2014
Published electronically: January 21, 2015
Additional Notes: The first author was supported by KOSEF-R01-2008-000-11553-0
The third author is a member of INdAM–GNAMPA
Communicated by: Tatiana Toro
Article copyright: © Copyright 2015 American Mathematical Society