The conormal derivative problem for higher order elliptic systems with irregular coefficients

TL;DR

This paper establishes $L_p$ estimates for solutions to higher-order elliptic systems with irregular coefficients under conormal boundary conditions, extending known results to more complex systems and less regular coefficients.

Contribution

It provides the first $L_p$ estimates for higher-order elliptic systems with irregular coefficients under conormal boundary conditions, even in the second-order case.

Findings

Established $L_p$ estimates for higher-order elliptic systems

Extended results to Reifenberg flat domains and irregular coefficients

Achieved new results even for second-order elliptic systems

Abstract

We prove $L_p$ estimates of solutions to a conormal derivative problem for divergence form complex-valued higher-order elliptic systems on a half space and on a Reifenberg flat domain. The leading coefficients are assumed to be merely measurable in one direction and have small mean oscillations in the orthogonal directions on each small ball. Our results are new even in the second-order case. The corresponding results for the Dirichlet problem were obtained recently in [15].