A second order approach to the Kato square root problem on open sets

TL;DR

This paper proves the Kato square root property for coupled second-order elliptic systems on open sets with mixed boundary conditions, using a second-order approach that simplifies and shortens the proof compared to previous methods.

Contribution

It introduces a second-order method to establish the Kato square root property for elliptic systems on open sets with mixed boundary conditions, avoiding the first-order approach.

Findings

Proves Kato square root property for coupled elliptic systems.

Uses geometric conditions like Ahlfors–David regularity and connectivity.

Provides a shorter, less complex proof compared to earlier work.

Abstract

We obtain the Kato square root property for coupled second-order elliptic systems in divergence form subject to mixed boundary conditions on an open and possibly unbounded set in $\mathbb{R}^n$ under two simple geometric conditions: The Dirichlet boundary parts for the respective components are Ahlfors--David regular and a quantitative connectivity property in the spirit of locally uniform domains holds near the remaining Neumann boundary parts. In contrast to earlier work, our proof is not based on the first-order approach due to Axelsson--Keith--McIntosh but uses a second-order approach in the spirit of the original solution to the Kato square root problem on Euclidean space. This way, the proof becomes substantially shorter and technically less demanding.