  In order to understand how screening is modified by electronic interferences
in a mesoscopic isolated system, we have computed both analytically and
numerically the average thermodynamic and time dependent polarisabilities of
two dimensional mesoscopic samples in the presence of an Aharonov-Bohm flux.
Two geometries have been considered: rings and squares. Mesoscopic correction
to screening are taken into account in a self consistent way, using the
response function formalism. The role of the statistical ensemble (canonical
and grand canonical), disorder and frequency have been investigated. We have
also computed first order corrections to the polarisability due to
electron-electron interactions. Our main results concern the diffusive regime.
In the canonical ensemble, there is no flux dependence polarisability when the
frequency is smaller than the level spacing. On the other hand, in the grand
canonical ensemble for frequencies larger than the mean broadening of the
energy levels (but still small compared to the level spacing), the
polarisability oscillates with flux, with the periodicity $h/2e$. The order of
magnitude of the effect is given by $\delta \alpha/\alpha \propto
(\lambda_s/Wg)$, where $\lambda$ is the Thomas Fermi screening length, $W$ the
width of the rings or the size of the squares and $g$ their average
dimensionless conductance. This magnetopolarisability of Aharonov-Bohm rings
has been recently measured experimentally \cite{PRL_deblock00} and is in good
agreement with our grand canonical result.
