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arXiv · 1712.09851

Lp-estimates for the square root of elliptic systems with mixed boundary conditions

Moritz Egert
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Abstract

This article focuses on Lp-estimates for the square root of elliptic systems of second order in divergence form on a bounded domain. We treat complex bounded measurable coefficients and allow for mixed Dirichlet/Neumann boundary conditions on domains beyond the Lipschitz class. If there is an associated bounded semigroup on Lp0 , then we prove that the square root extends for all p $\in$ (p0, 2) to an isomorphism between a closed subspace of W1p carrying the boundary conditions and Lp. This result is sharp and extrapolates to exponents slightly above 2. As a byproduct, we obtain an optimal p-interval for the bounded H$\infty$-calculus on Lp. Estimates depend holomorphically on the coefficients, thereby making them applicable to questions of non-autonomous maximal regularity and optimal control. For completeness we also provide a short summary on the Kato square root problem in L2 for systems with lower order terms in our setting.

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Moritz Egert. 2017-12-28. Lp-estimates for the square root of elliptic systems with mixed boundary conditions. https://doi.org/10.1016/j.jde.2018.04.002

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