Curvature effects in pattern formation: well-posedness and optimal control of a sixth-order Cahn-Hilliard equation

Pierluigi Colli; Gianni Gilardi; Andrea Signori; Jürgen Sprekels

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arXiv · 2401.05189

Curvature effects in pattern formation: well-posedness and optimal control of a sixth-order Cahn-Hilliard equation

Pierluigi Colli·Gianni Gilardi·Andrea Signori·Jürgen Sprekels

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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 improve 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.

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Pierluigi Colli, Gianni Gilardi, Andrea Signori, Jürgen Sprekels. 2024-01-10. Curvature effects in pattern formation: well-posedness and optimal control of a sixth-order Cahn-Hilliard equation. https://arxiv.org/abs/2401.05189

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