Weak solutions and sharp interface limit of the anisotropic Cahn-Hilliard equation with disparate mobility and inhomogeneous potential

We study the existence of weak solutions and the corresponding sharp interface limit of an anisotropic Cahn-Hilliard equation with disparate mobility, i.e., the mobility is degenerate in one of the two pure phases, making the diffusion in that phase vanish. The double-well potential is polynomial and is weighted by a spatially inhomogeneous coefficient. In the limit when the parameter of the interface width tends to zero, and under an energy convergence assumption, we prove that the weak solutions converge to BV solutions of a weighted anisotropic Hele-Shaw flow. We also add some numerical simulations to analyze the effects of anisotropy on the Cahn-Hilliard equation.