  In this paper we consider the equilibrium problem in the relaxed linear model
of micromorphic elastic materials. The basic kinematical fields of this
extended continuum model are the displacement $u\in \mathbb{R}^3$ and the
non-symmetric micro-distortion density tensor $P\in \mathbb{R}^{3\times 3}$. In
this relaxed theory a symmetric force-stress tensor arises despite the presence
of microstructure and the curvature contribution depends solely on the
micro-dislocation tensor ${\rm Curl}\, P$. However, the relaxed model is able
to fully describe rotations of the microstructure and to predict non-polar
size-effects. In contrast to classical linear micromorphic models, we allow the
usual elasticity tensors to become positive-semidefinite. We prove that,
nevertheless, the equilibrium problem has a unique weak solution in a suitable
Hilbert space. The mathematical framework also settles the question of which
boundary conditions to take for the micro-distortion. Similarities and
differences between linear micromorphic elasticity and dislocation gauge theory
are discussed and pointed out.
