Abstract
We employ a quantum generalization of the self-consistent harmonic
approximation to investigate the role of grain boundaries
and of $^3$He impurities in solid $^4$He. We show that,
under the experimental circumstances, surfaces melting
and quantum lattice fluctuations make grain boundaries
and $^3$He impurities always unstable against formation of
$^4$He liquid phases. We show also that the anomalous dependence of the
critical temperature and of the magnitude of the non classical
rotational inertia is compatible with a simple model where
$^3$He impurities rule the size of the grain boundaries.
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