Extension to unbounded non-convex smooth domains

Extend the fibred JKO convergence and solution-existence results for Cahn–Hilliard-type energies to non-convex smooth domains that are unbounded.

Background

The paper establishes variational convergence and existence of dissipative or energy-dissipating solutions in two spatial settings: the whole space $\mathbb{R}^d$ and smooth bounded domains, including non-convex domains equipped with the intrinsic length distance. The authors identify the combination of non-convex geometry and an unbounded spatial domain as a setting not covered by their analysis.

An extension to unbounded non-convex smooth domains would require combining the difficulties associated with intrinsic transport on non-convex geometries and the compactness and control-at-infinity issues arising on unbounded domains.