  In the theory of non-linear parabolic and elliptic partial differential
equations, the notion of maximal regularity plays an essential role in
establishing existence, regularity and boundedness of solutions. There is a
long history of works where sufficient conditions for maximal regularity have
been established: First scalar equations and systems of finitely many coupled
equations have been considered. Around 2000, the vector-valued case with
infinite-dimensional range space $E$ became accessible to the development and
progress in theory of $\mathcal{R}$-bounded operator families and its close
connection to the $\mathcal{H}^\infty$-calculus. The ground-braking results by
Denk, Hieber and Pr\"uss for $\mathrm{L}_p$-maximal regularity of vector-valued
parabolic and elliptic boundary value problems, however, were restricted to
boundary conditions with homogeneous principle parts of the boundary symbol, in
contrast to some previous results by Ladyszenskaya, Solonnikov and Uralceva for
finite-component systems which also allow for, e.g. both Dirichlet and (mixed)
flux boundary conditions at the same position. In this manuscript we aim for
closing this gap, and extend the results of Denk, Hieber and Pr\"uss to this
slightly more general situation. To this end, we closely review the strategy
used in their works and adapt it to the situation considered here.
