Weighted distribution approach to gradient estimates for quasilinear elliptic double-obstacle problems in Orlicz spaces

TL;DR

This paper develops a weighted distribution method to obtain global regularity estimates for elliptic double-obstacle problems in Orlicz spaces, extending previous results using fractional maximal distributions and analyzing gradient estimates via fractional operators.

Contribution

It introduces a novel weighted distribution approach for regularity estimates in Orlicz spaces and extends existing results by incorporating fractional maximal distributions and weak BMO conditions.

Findings

Established global regularity estimates in Orlicz and Lorentz spaces.

Extended regularity results using weighted fractional maximal distributions.

Analyzed gradient estimates via fractional maximal operators and Riesz potentials.

Abstract

We construct an efficient approach to deal with the global regularity estimates for a class of elliptic double-obstacle problems in Lorentz and Orlicz spaces. The motivation of this paper comes from the study on an abstract result in the viewpoint of the fractional maximal distributions and this work also extends some regularity results proved in \cite{PN_dist} by using the weighted fractional maximal distributions (WFMDs). We further investigate a pointwise estimates of the gradient of weak solutions via fractional maximal operators and Riesz potential of data. Moreover, in the setting of the paper, we are led to the study of problems with nonlinearity is supposed to be partially weak BMO condition (is measurable in one fixed variable and only satisfies locally small-BMO seminorms in the remaining variables).