  We consider the classical Mindlin-Eringen linear micromorphic model with a
new strictly weaker set of displacement boundary conditions. The new consistent
coupling condition aims at minimizing spurious influences from arbitrary
boundary prescription for the additional microdistortion field P. In effect, P
is now only required to match the tangential derivative of the classical
displacement u which is known at the Dirichlet-part of the boundary. We derive
the full boundary condition, in adding the missing Neumann condition on the
Dirichlet-part. We show existence and uniqueness of the static problem for this
weaker boundary condition. These results are based on new coercive inequalities
for incompatible tensor fields with prescribed tangential part. Finally, we
show that compared to classical Dirichlet conditions on u and P, the new
boundary condition modifies the interpretation of the constitutive parameters.
