Curvature effects in pattern formation: well-posedness and optimal control of a sixth-order Cahn-Hilliard equation
arxiv(2024)
摘要
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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