WIAS Preprint No. 3085, (2024)

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


  • 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


  • Sixth-order Cahn--Hilliard equation, functionalized Cahn--Hilliard equation, Willmore regularization, curvature effects, well-posedness, optimal control, first-order necessary optimality conditions




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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