  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 $\frac{1}{2}$ 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.
