  In this paper, we study the sharp interface limit for solutions of the
Cahn-Hilliard equation with disparate mobilities. This means that the mobility
function degenerates in one of the two energetically favorable configurations,
suppressing the diffusion in that phase. First, we construct suitable weak
solutions to this Cahn-Hilliard equation. Second, we prove precompactness of
these solutions under natural assumptions on the initial data. Third, under an
additional energy convergence assumption, we show that the sharp interface
limit is a distributional solution to the Hele-Shaw flow with optimal
energy-dissipation rate.
