Elliptic equations with singular BMO coefficients in Reifenberg domains

TL;DR

This paper establishes uniform $W^{1,p}$ estimates for elliptic equations with coefficients having large jumps in Reifenberg flat domains, extending regularity results to more irregular boundary conditions.

Contribution

It provides new $W^{1,p}$ estimates for elliptic equations with BMO coefficients in Reifenberg domains, including cases with large jumps along subdomain boundaries.

Findings

Uniform $W^{1,p}$ estimates are achieved in Reifenberg flat domains.

Estimates hold even with large jumps in coefficients along subdomain boundaries.

Results extend regularity theory to more irregular boundary and coefficient conditions.

Abstract

$W^{1, p}$ estimate for the solutions of elliptic equations whose coefficient matrix can have large jump along the boundary of subdomains is obtained. The principal coefficients are supposed to be in the John-Nirenberg space with small BMO seminorms. The domain and subdomains are Reifenberg flat domains and moreover, it has been shown that the estimates are uniform with respect to the distance between the subdomains.