Curvature effects in pattern formation: Well-posedness and optimal control of a sixth-order Cahn--Hilliard equation
Authors
- Colli, Pierluigi
ORCID: 0000-0002-7921-5041 - Gilardi, Gianni
ORCID: 0000-0002-0651-4307 - Signori, Andrea
ORCID: 0000-0001-7025-977X - Sprekels, Jürgen
ORCID: 0009-0000-0618-8604
2020 Mathematics Subject Classification
- 35K55 35K51 49J20 49K20 49J50
Keywords
- Sixth-order Cahn--Hilliard equation, functionalized Cahn--Hilliard equation, Willmore regularization, curvature effects, well-posedness, optimal control, first-order necessary optimality conditions
DOI
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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