Curvature Effects in Pattern Formation: Well-Posedness and Optimal Control of a Sixth-Order Cahn–Hilliard Equation

SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4928-4969, August 2024.
Abstract. This work investigates the well-posedness and optimal control of a sixth-order Cahn–Hilliard equation, a higher-order variant of the celebrated and well-established Cahn–Hilliard equation. The equation is endowed with a source term, where the control variable enters as a distributed mass regulator. The inclusion of additional spatial derivatives in the sixth-order formulation enables the model to capture curvature effects, leading to a more accurate depiction of isothermal phase separation dynamics in complex materials systems. We provide a well-posedness result for the aforementioned system when the corresponding nonlinearity of double-well shape is regular and then analyze a corresponding optimal control problem. For the latter, existence of optimal controls is established, and the first-order necessary optimality conditions are characterized via a suitable variational inequality. These results aim at contributing to improving the understanding of the mathematical properties and control aspects of the sixth-order Cahn–Hilliard equation, offering potential applications in the design and optimization of materials with tailored microstructures and properties.