Titlebook: Square Roots of Elliptic Systems in Locally Uniform Domains; Sebastian Bechtel Book 2024 The Editor(s) (if applicable) and The Author(s),

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First Order Formalism,We introduce auxiliary first-order differential operators that are related to our elliptic system. This follows the ideas of Axelsson–Keith–McIntosh. The main result are quadratic estimates for the first-order system. We will use them in the next chapter to show the Kato square root property on interior thick sets.

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,Kato’s Square Root Property on Thick Sets,We are now in a position to prove Kato’s square root property on locally uniform domains but still requiring the interior thickness condition.

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Removing the Thickness Condition,The result that we are going to show in this chapter is notably the main result of this monograph. In fact, it reads almost the same as Theorem . from the foregoing chapter, despite that we are able to eliminate the assumption that . is interior thick. Consequently, we are going to write . instead of ..

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Sebastian BechtelProvides a complete framework to treat elliptic and parabolic problems incorporating mixed boundary conditions.Introduces global approaches to cover problems subject to mixed boundary conditions.Solve

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A Short Account on Sectorial and Bisectorial Operators,in divergence form (sectorial operators), which we will introduce in the subsequent Chap. ., and their associated first-order formulations (bisectorial operators), which we study in more detail in Chap. .. Functional calculus theory will be heavily and freely used in large parts of this book.