  We present in this paper a detailed analysis of the flexoelectric instability
of a planar nematic layer in the presence of an alternating electric field
(frequency $\omega$), which leads to stripe patterns (flexodomains) in the
plane of the layer. This equilibrium transition is governed by the free energy
of the nematic which describes the elasticity with respects to the
orientational degrees of freedom supplemented by an electric part. Surprisingly
the limit $\omega \to 0$ is highly singular. In distinct contrast to the
dc-case, where the patterns are stationary and time-independent, they appear at
finite, small $\omega$ periodically in time as sudden bursts. Flexodomains are
in competition with the intensively studied electro-hydrodynamic instability in
nematics, which presents a non-equilibrium dissipative transition. It will be
demonstrated that $\omega$ is a very convenient control parameter to tune
between flexodomains and convection patterns, which are clearly distinguished
by the orientation of their stripes.
