Global weighted estimates for nonlinear elliptic obstacle problems over Reifenberg domains

Byun, Sun-Sig; Cho, Yumi; Palagachev, Dian

doi:10.1090/S0002-9939-2015-12458-6

Global weighted estimates for nonlinear elliptic obstacle problems over Reifenberg domains
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by Sun-Sig Byun, Yumi Cho and Dian K. Palagachev PDF

Proc. Amer. Math. Soc. 143 (2015), 2527-2541 Request permission

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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