Curvature effects in pattern formation: well-posedness and optimal control of a sixth-order Cahn-Hilliard equation
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914776220696576 |
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| author | Colli, Pierluigi Gilardi, Gianni Signori, Andrea Sprekels, Jürgen |
| author_facet | Colli, Pierluigi Gilardi, Gianni Signori, Andrea Sprekels, Jürgen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_05189 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Curvature effects in pattern formation: well-posedness and optimal control of a sixth-order Cahn-Hilliard equation Colli, Pierluigi Gilardi, Gianni Signori, Andrea Sprekels, Jürgen Analysis of PDEs Optimization and Control 35K55, 35K51, 49J20, 49K20, 49J50 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. |
| title | Curvature effects in pattern formation: well-posedness and optimal control of a sixth-order Cahn-Hilliard equation |
| topic | Analysis of PDEs Optimization and Control 35K55, 35K51, 49J20, 49K20, 49J50 |
| url | https://arxiv.org/abs/2401.05189 |