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We consider the negative Laplacian subject to mixed boundary conditions on abounded domain. We prove under very general geometric assumptions that slightlyabove the critical exponent $\frac{1}{2}$ its fractional power domains stillcoincide with suitable Sobolev spaces of optimal regularity. In combinationwith a reduction theorem recently obtained by the authors, this solves the KatoSquare Root Problem for elliptic second order operators and systems indivergence form under the same geometric assumptions.

The Kato Square Root Problem for Mixed Boundary Conditions