Global solution to the Allen–Cahn equation with singular potentials and dynamic boundary conditions

Abstract

Abstract. We prove well-posedness results for the solution to an initial and boundaryvalue problem for an Allen-Cahn type equation describing the phenomenon of phase transitions for a material contained in a bounded and regular domain. The dynamic boundary conditions for the order parameter have been recently proposed by some physicists to account for interactions with the walls. We show our results using suitable regularizations of the nonlinearities of the problem and performing some a priori estimates which allow us to pass to the limit thanks to compactness and monotonicity arguments.---------------------

Recent Results on the Cahn - Hilliard Equation with Dynamic Boundary Conditions

“…Roughly speaking, condition (1.6) is opposite to the one postulated in [11]. On the contrary, it is the same as the one introduced in the paper [18], which however deals with the Allen Cahn equation.…”

“…In this note, we follow a strategy developed in [18] to investigate the Allen Cahn equation with dynamic boundary conditions, which consists in letting f Γ be the leading potential with respect to f : it turns out that this approach simplies the analysis. Moreover, we discuss the optimal boundary control problem for the viscous and pure Cahn…”

“…Our problem consists in looking for a quintuplet (y, y Γ , w, ξ, ξ Γ ) such that 18) and satisfying for a.a. t ∈ (0, T ) the variational equations …”

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).…”

A boundary control problem for the pure Cahn–Hilliard equation with dynamic boundary conditions

“…for some positive constants η and C and for every r ∈ D. This condition, earlier introduced in [4] in relation with the Allen-Cahn equation with dynamic boundary conditions (see also [11]), is then used in [9] (as well as in [6] and [10]) to deal with the Cahn-Hilliard system. This complements [14], where some kind of an opposite inequality is assumed.…”