The Regularity problem in domains with lower dimensional boundaries

TL;DR

This paper proves the solvability of the Regularity boundary value problem in domains with lower dimensional boundaries for certain elliptic operators, introducing new estimates to handle multiple non-tangential derivatives.

Contribution

It establishes solvability results for the Regularity problem in lower dimensional boundary domains with small oscillation coefficients, using novel estimates for angular derivatives.

Findings

Solvability of the Regularity boundary value problem in lower dimensional domains.

Development of new estimates on angular derivatives for multiple non-tangential derivatives.

Extension of classical methods to settings with lower dimensional boundaries.

Abstract

In the present paper we establish the solvability of the Regularity boundary value problem in domains with (flat and Lipschitz) lower dimensional boundaries for operators whose coefficients exhibit small oscillations analogous to the Dahlberg-Kenig-Pipher condition. The proof follows the classical strategy of showing bounds on the square function and the non-tangential maximal function. The key novelty and difficulty of this setting is the presence of multiple non-tangential derivatives. To solve it, we consider a cylindrical system of derivatives and establish new estimates on the "angular derivatives".