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Elliptic obstacle problems with measurable nonlinearities in non-smooth domains

유승진|김유찬
대한수학회 (The Korean Mathematical Society)|2019.01|대한수학회지 (Journal of the KMS)|제56권 제1호|pp. 239-263|DOI 10.4134/JKMS.j180157

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키워드

BMOCalder\'{o}n-Zygmund type estimatenonlinear elliptic obstacle problemmeasurable nonlinearityReifenberg flat domain
주제분류수학

초록

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 $\ba(\xi,x_1,x')$ 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.

Abstract

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 $\ba(\xi,x_1,x')$ 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.

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