  Within the Landau-de Gennes theory of liquid crystals, we study theoretically
the equilibrium configurations with uniaxial symmetry. For an arbitrary form of
the bulk energy density, we show that energy minimizers among uniaxially
symmetric configurations are usually not equilibrium configurations. It was
known before that uniaxiality can sometimes be broken by energy minimizers. We
prove here that this is always the case in one or two dimensions. Even more we
prove that equilibrium configurations (not only minimizers) cannot be uniaxial
in one or two dimensions (unless they are constant). Namely, uniaxial
equilibrium configurations with at least one translational invariance direction
must have a uniform director field. In the case of a hybrid nematic cell, or of
a capillary with radial anchoring, our result implies that biaxial escape
always occurs, and that it does so not only in the core of a defect: uniaxial
order is destroyed in the whole cell.
