Optimal regularity for a Dirichlet-conormal problem in Reifenberg flat domain

TL;DR

This paper establishes the optimal range of p for which divergence form elliptic equations with mixed boundary conditions are solvable in Reifenberg flat domains, under small oscillation assumptions.

Contribution

It proves the unique solvability of second-order elliptic equations with mixed boundary conditions in the optimal p-range in Reifenberg flat domains, with minimal regularity assumptions.

Findings

Unique $W^{1,p}$ solvability for $p ext{ in } (4/3,4)$

Solvability under small mean oscillation of coefficients

Results applicable to Reifenberg flat domains with separated boundary conditions

Abstract

We study the divergence form second-order elliptic equations with mixed Dirichlet-conormal boundary conditions. The unique $W^{1,p}$ solvability is obtained with $p$ being in the optimal range $(4/3,4)$. The leading coefficients are assumed to have small mean oscillations and the boundary of domain is Reifenberg flat. We also assume that the two boundary conditions are separated by some Reifenberg flat set of co-dimension $2$ on the boundary.