Abstract. We consider the negative Laplacian subject to mixed boundary conditions on a bounded domain. We prove under very general geometric assumptions that slightly above the critical exponent 1 2 its fractional power domains still coincide with suitable Sobolev spaces of optimal regularity. In combination with a reduction theorem recently obtained by the authors, this solves the Kato Square Root Problem for elliptic second order operators and systems in divergence form under the same geometric assumptions. Thereby we answer a question posed by J. L. Lions in 1962 [29].

Abstract. On a domain Ω ⊆ R d we consider second order elliptic systems in divergence form with bounded complex coefficients, realized via a sesquilinear form with domain V ⊆ H 1 (Ω). Under very mild assumptions on Ω and V we show that the solution to the Kato Square Root Problem for such systems can be deduced from a regularity result for the fractional powers of the negative Laplacian in the same geometric setting. This extends an earlier result of M c Intosh [24] to non-smooth coefficients.

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On bounded domains Ω ⊂ R d , d ≥ 2, reaching far beyond the scope of Lipschitz domains, we consider an elliptic system of order 2m in divergence form with complex L ∞coefficients complemented with homogeneous mixed Dirichlet/Neumann boundary conditions. We prove that the L p -realization of the corresponding operatorTo perform this proof, we generalize the L p -extrapolation theorem of Shen to the Banach space valued setting and to arbitrary Lebesgue-measurable underlying sets. Notation, assumptions and preliminary considerationsThroughout this article the space dimension d ≥ 2 is fixed. An open and connected subset of R d will be called a domain. A ball with center x and radius r is denoted by B(x, r), whereas a cube centered at x, with diameter 2r, and faces parallel to the coordinate axes is denoted by Q(x, r). For a positive constant α denote the dilated balls with same center by αB. Integration

In this article, an L1 maximal regularity theory for parabolic evolution equations inspired by the pioneering work of Da Prato and Grisvard [22] is developed. Besides of its own interest, the approach yields a framework allowing global-in-time control of the change of Eulerian to Lagrangian coordinates in various problems related to fluid mechanics. This property is of course decisive for free boundary problems. As an application, a global well-posedness result for the free boundary value problem for incompressible Navier-Stokes equations is established in the case where the initial domain coincides with the half-space, and the initial velocity is small with respect to a suitable scaling invariant norm. ContentsChapter 1. Introduction iii Chapter 2. The Da Prato-Grisvard theorem Chapter 3. The functional setting and basic interpolation results Chapter 4. The Stokes operator with a Neumann-type boundary condition