Boundary control problem and optimality conditions for the Cahn–Hilliard equation with dynamic boundary conditions

Colli, Pierluigi; Signori, Andrea

doi:10.1080/00207179.2019.1680870

Cited by 5 publications

(2 citation statements)

References 42 publications

(53 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A very general approach for distributed control problems for possibly fractional equations of CH-type is carried out in the papers [20,21,22], with an extension of the analysis to double obstacle potentials like f 2obs in (1.7) via deep quench approximation. The coupling of CH equations in the bulk with dynamic boundary conditions has been investigated in [14,15,23], and the presence of a convective term with the velocity vector taken as control has been dealt with in [16,17,19,30] (see also the references in the quoted contributions).…”

Section: Introductionmentioning

confidence: 99%

Well-posedness and optimal control for a Cahn–Hilliard–Oono system with control in the mass term

Colli

Gilardi

Rocca

et al. 2022

DCDS-S

Self Cite

View full text Add to dashboard Cite

<p style='text-indent:20px;'>The paper treats the problem of optimal distributed control of a Cahn–Hilliard–Oono system in <inline-formula><tex-math id="M1">\begin{document}$ {{\mathbb{R}}}^d $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ 1\leq d\leq 3 $\end{document}</tex-math></inline-formula>, with the control located in the mass term and admitting general potentials that include both the case of a regular potential and the case of some singular potential. The first part of the paper is concerned with the dependence of the phase variable on the control variable. For this purpose, suitable regularity and continuous dependence results are shown. In particular, in the case of a logarithmic potential, we need to prove an ad hoc strict separation property, and for this reason we have to restrict ourselves to the case <inline-formula><tex-math id="M3">\begin{document}$ d = 2 $\end{document}</tex-math></inline-formula>. In the rest of the work, we study the necessary first-order optimality conditions, which are proved under suitable compatibility conditions on the initial datum of the phase variable and the time derivative of the control, at least in case of potentials having unbounded domain.</p>

show abstract

Section: Introductionmentioning

confidence: 99%

Well-posedness and optimal control for a Cahn–Hilliard–Oono system with control in the mass term

Colli

Gilardi

Rocca

et al. 2022

DCDS-S

Self Cite

View full text Add to dashboard Cite

show abstract

“…The articles [12,15,18] refer instead to similar approaches but for different equations in the domain. Let us also mention the papers [17,21,24] devoted to the analysis of optimal control problems for some Cahn-Hilliard systems coupling equation and dynamic boundary condition. For completeness, let us also mention that vanishing diffusion studies on Cahn-Hilliard and Allen-Cahn equations have been pursued also in the case of stochastic forcing, for which we refer to [39] and [36], respectively.…”

Section: Introductionmentioning

confidence: 99%

The Cahn-Hilliard equation with forward-backward dynamic boundary condition via vanishing viscosity

Colli¹,

Fukao²,

Scarpa³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

An asymptotic analysis for a system with equation and dynamic boundary condition of Cahn-Hilliard type is carried out as the coefficient of the surface diffusion acting on the phase variable tends to 0, thus obtaining a forward-backward dynamic boundary condition at the limit. This is done in a very general setting, with nonlinear terms admitting maximal monotone graphs both in the bulk and on the boundary. The two graphs are related by a growth condition, with the boundary graph that dominates the other one. It turns out that in the limiting procedure the solution of the problem looses some regularity and the limit equation has to be properly interpreted in the sense of a subdifferential inclusion. However, the limit problem is still well-posed since a continuous dependence estimate can be proved. Moreover, in the case when the two graphs exhibit the same growth, it is shown that the solution enjoys more regularity and the boundary condition holds almost everywhere. An error estimate can also be shown, for a suitable order of the diffusion parameter.

show abstract

The Cahn--Hilliard Equation with Forward-Backward Dynamic Boundary Condition via Vanishing Viscosity

Colli¹,

Fukao²,

Scarpa

2022

SIAM J. Math. Anal.

Self Cite

View full text Add to dashboard Cite

A system with equation and dynamic boundary condition of Cahn-Hilliard type is considered. This system comes from a derivation performed in Liu-Wu (Arch. Ration. Mech. Anal. 233 (2019), 167-247) via an energetic variational approach. Actually, the related problem can be seen as a transmission problem for the phase variable in the bulk and the corresponding variable on the boundary. The asymptotic behavior as the coefficient of the surface diffusion acting on the boundary phase variable goes to 0 is investigated. By this analysis we obtain a forward-backward dynamic boundary condition at the limit. We can deal with a general class of potentials having a double-well structure, including the non-smooth double-obstacle potential. We illustrate that the limit problem is well-posed by also proving a continuous dependence estimate. Moreover, in the case when the two graphs, in the bulk and on the boundary, exhibit the same growth, we show that the solution of the limit problem is more regular and we prove an error estimate for a suitable order of the diffusion parameter.

show abstract

Boundary control problem and optimality conditions for the Cahn–Hilliard equation with dynamic boundary conditions

Cited by 5 publications

References 42 publications

Well-posedness and optimal control for a Cahn–Hilliard–Oono system with control in the mass term

Well-posedness and optimal control for a Cahn–Hilliard–Oono system with control in the mass term

The Cahn-Hilliard equation with forward-backward dynamic boundary condition via vanishing viscosity

The Cahn--Hilliard Equation with Forward-Backward Dynamic Boundary Condition via Vanishing Viscosity

Contact Info

Product

Resources

About