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arXiv · 2406.12812

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

Sebastian Bechtel·Cody Hutcheson·Tim Schmatzler·Tolgahan Tasci·Mattes Wittig
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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.

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Sebastian Bechtel, Cody Hutcheson, Tim Schmatzler, Tolgahan Tasci, Mattes Wittig. 2024-06-18. A second order approach to the Kato square root problem on open sets. https://arxiv.org/abs/2406.12812

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