  We develop a maximal regularity approach in temporally weighted $L_p$-spaces
for vector-valued parabolic initial-boundary value problems with inhomogeneous
boundary conditions, both of static and of relaxation type. Normal ellipticity
and conditions of Lopatinskii-Shapiro type are the basic structural
assumptions. The weighted framework allows to reduce the initial regularity and
to avoid compatibility conditions at the boundary, and it provides an inherent
smoothing effect of the solutions. Our main tools are interpolation and trace
theory for anisotropic Slobodetskii spaces with temporal weights,
operator-valued functional calculus, as well as localization and perturbation
arguments.
