The Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions is considered and well-posedness results are proved.

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Abstract. We introduce a fractional variant of the Cahn-Hilliard equation settled in a bounded domain Ω ⊂ R N and complemented with homogeneous Dirichlet boundary conditions of solid type (i.e., imposed in the whole of R N \ Ω). After setting a proper functional framework, we prove existence and uniqueness of weak solutions to the related initial-boundary value problem. Then, we investigate some significant singular limits obtained as the order of either of the fractional Laplacians appearing in the equation is let tend to 0. In particular, we can rigorously prove that the fractional Allen-Cahn, fractional porous medium, and fractional fast-diffusion equations can be obtained in the limit. Finally, in the last part of the paper, we discuss existence and qualitative properties of stationary solutions of our problem and of its singular limits.

In this paper, a distributed optimal control problem is studied for a diffuse interface model of tumor growth which was proposed by Hawkins-Daruud et al. in [25]. The model consists of a Cahn-Hilliard equation for the tumor cell fraction ϕ coupled to a reaction-diffusion equation for a function σ representing the nutrient-rich extracellular water volume fraction. The distributed control u monitors as a right-hand side the equation for σ and can be interpreted as a nutrient supply or a medication, while the cost function, which is of standard tracking type, is meant to keep the tumor cell fraction under control during the evolution. We show that the control-to-state operator is Fréchet differentiable between appropriate Banach spaces and derive the first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables.

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