  There is growing evidence, from experiments and numerical simulations, that a
key feature of sufficiently disordered superconductors is the spatial
inhomogeneity of the order parameter. However not much is known analytically
about the details of its spatial distribution or the associated global critical
temperature that signals the breaking of long-range order. Here we address this
problem for disordered systems around an Anderson transition characterized by
multifractal one-body eigenstates. In the limit of weak multifractality and for
weakly coupled superconductors we compute the superconducting order parameter
analytically, including its energy dependence and statistical distribution in
space. The spatial distribution of the order parameter is found to be always
log-normal. The global critical temperature, computed by percolation techniques
and neglecting phase fluctuations, is enhanced with respect to the clean limit
only for very weakly coupled superconductors. Some enhancement still persists
even in the presence of moderate phase fluctuations crudely modelled by
increasing the percolation threshold. Our results are also consistent with
experiments, where enhancement of the critical temperature is observed in Al
thin films, a very weakly coupled metallic superconductor, but not in more
strongly coupled materials.
