The Dirichlet-conormal problem with homogeneous and inhomogeneous boundary conditions

TL;DR

This paper investigates the regularity and boundary behavior of solutions to the mixed Dirichlet-conormal problem on irregular Reifenberg-flat domains, providing new estimates under various geometric conditions.

Contribution

It establishes $W^{1,p}$ regularity and non-tangential maximal function estimates for the problem on complex domains with mixed boundary conditions.

Findings

Proves $W^{1,p}$ regularity on Reifenberg-flat domains.

Establishes non-tangential maximal function estimates under Lipschitz domain assumptions.

Handles interfaces that are Reifenberg-flat or close to Lipschitz functions.

Abstract

We consider the mixed Dirichlet-conormal problem on irregular domains in $\mathbb{R}^d$. Two types of regularity results will be discussed: the $W^{1,p}$ regularity and a non-tangential maximal function estimate. The domain is assumed to be Reifenberg-flat, and the interfacial boundary is either Reifenberg-flat of co-dimension $2$ or is locally sufficiently close to a Lipschitz function of $m$ variables, where $m=1,\ldots,d-2$. For the non-tangential maximal function estimate, we also require the domain to be Lipschitz.