# ELLIPTIC OBSTACLE PROBLEMS WITH MEASURABLE NONLINEARITIES IN NON-SMOOTH DOMAINS

Kim, Youchan;Ryu, Seungjin

• Received : 2018.03.06
• Accepted : 2018.07.05
• Published : 2019.01.01
• 157 6

#### Abstract

The $Calder{\acute{o}}n$-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 $a({\xi},x_1,x^{\prime})$ 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

$Calder{\acute{o}}n$-Zygmund type estimate;nonlinear elliptic obstacle problem;measurable nonlinearity;BMO;Reifenberg flat domain

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#### Acknowledgement

Supported by : National Research Foundation of Korea