The Allen–Cahn equation with dynamic boundary conditions and mass constraints

Abstract

The Allen-Cahn equation, coupled with dynamic boundary conditions, has recently received a good deal of attention. The new issue of this paper is the setting of a rather general mass constraint which may involve either the solution inside the domain or its trace on the boundary. The system of nonlinear partial differential equations can be formulated as variational inequality. The presence of the constraint in the evolution process leads to additional terms in the equation and the boundary condition containing a suitable Lagrange multiplier. A well-posedness result is proved for the related initial value problem.

On a coupled bulk–surface Allen–Cahn system with an affine linear transmission condition and its approximation by a Robin boundary condition

“…Immediately one observes that there will be difficulties in passing to the limit in some approximation scheme to obtain the last term of (1.3) if g is a nonlinear function. In the case g (and also h) is an affine linear function, i.e., g(s) = α −1 (s − η) for some α ≠ 0, η ∈ R, we can appeal to the procedure in Colli and Fukao [14] to deduce the existence of strong solutions to (1.2). However, for a more general and possibly nonlinear relation even the existence of a weak solution to (1.2) is an open problem due to the highly nonlinear surface equation.…”

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

“…In this regard, let us address to [19], where both the viscous and the non-viscous Cahn-Hilliard equations, combined with these kinds of boundary conditions, have been investigated by assuming the boundary potential to be dominant on the bulk one. Furthermore, we have to mention [4,9,13,16,23,25,33,[36][37][38]42], where other problems related to the Cahn-Hilliard equation combined with dynamic boundary conditions have been analyzed, and [3,7,8,11,20,29,35] for the coupling of dynamic boundary conditions with different phase field models such as the Allen-Cahn or the Penrose-Fife model. So, according to [19] we supply the above system (1.1)-(1.2) with ∂ n w = 0 on Σ := Γ × (0, T ), (1.5) ∂ n y + ∂ t y Γ − ∆ Γ y Γ + f ′ Γ (y Γ ) = u Γ on Σ, (1.6) where Γ is the boundary of Ω, y Γ denotes the trace of y, ∆ Γ stands for the Laplace-Beltrami operator on the boundary, and ∂ n represents the outward normal derivative.…”

A Boundary Control Problem for the Viscous Cahn–Hilliard Equation with Dynamic Boundary Conditions

“…In the former, the reader can find the physical meaning and free energy derivation of the boundary value problem given by (1.1) and (1.4)-(1.5), besides the mathematical treatment of the problem itself. The latter provides existence, uniqueness and regularity results for the same boundary value problem by assuming that the dominating potential is the boundary potential W Γ rather than the bulk potential W (thus, in contrast to [14]) and thus it is close from this point of view to [4], where the Allen-Cahn equation with dynamic boundary condition is studied (see also [7] in which a mass constraint is considered, too).…”