A L^infty bound for the Cahn--Hilliard equation with relaxed non-smooth free energy

Phase field models are widely used to describe multiphase systems. Here a smooth indicator function, called phase field, is used to describe the spatial distribution of the phases under investigation. Material properties like density or viscosity are introduced as given functions of the phase field. These parameters typically have physical bounds to fulfil, e.g. positivity of the density. To guarantee these properties, uniform bounds on the phase field are of interest. In this work we derive a uniform bound on the solution of the Cahn--Hilliard system, where we use the double-obstacle free energy, that is relaxed by Moreau--Yosida relaxation.