Journal of the
Korean Mathematical Society JKMS

The Calder\'{o}n-Zygmund type estimate is proved for elliptic obstacle problems in bounded non-smooth domains. The problems are related to divergence form nonlinear elliptic equation with measurable nonlinearities. Precisely, nonlinearity $\mathbf{a}(\xi,x_1,x')$ is assumed to be only measurable in one spatial variable $x_1$ and has locally small BMO semi-norm in the other spatial variables $x'$, uniformly in $\xi$ variable. Regarding non-smooth domains, we assume that the boundaries are locally flat in the sense of Reifenberg. We also investigate global regularity in the settings of weighted Orlicz spaces for the weak solutions to the problems considered here.

Keywords: Calder\'{o}n-Zygmund type estimate, nonlinear elliptic obstacle problem, measurable nonlinearity, BMO, Reifenberg flat domain

MSC numbers: Primary 35J87; Secondary 35R05

Supported by: The first author was supported by the National Research Foundation of Korea(NRF-2017R1D1A1B03034302) grant funded by the Korea government. The second author was supported by the 2015 Research Fund of the University of Seoul.