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

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 a 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. Copyright © 2014 John Wiley & Sons, Ltd.

[1]  Alain Miranville,et al.  A Variational Approach to a Cahn–Hilliard Model in a Domain with Nonpermeable Walls , 2013 .

[2]  Alain Miranville,et al.  ATTRACTORS FOR A CAGINALP MODEL WITH A LOGARITHMIC POTENTIAL AND COUPLED DYNAMIC BOUNDARY CONDITIONS , 2013 .

[3]  Nobuyuki Kenmochi,et al.  CHAPTER 4 - Monotonicity and Compactness Methods for Nonlinear Variational Inequalities , 2007 .

[4]  Periodic stability of solutions to some degenerate parabolic equations with dynamic boundary conditions , 1996 .

[5]  Matthias Liero,et al.  Passing from bulk to bulk-surface evolution in the Allen–Cahn equation , 2013 .

[6]  M. Kubo.  The Cahn–Hilliard equation with time-dependent constraint , 2012 .

[7]  A. Grigor’yan.  Heat Kernel and Analysis on Manifolds , 2012 .

[8]  M. Warma,et al.  Well posedness and the global attractor of some quasi-linear parabolic equations with nonlinear dynamic boundary conditions , 2010, Differential and Integral Equations.

[9]  A. Miranville,et al.  A Cahn-Hilliard-Gurtin model with dynamic boundary conditions , 2012 .

[10]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[11]  Viorel Barbu,et al.  Nonlinear Differential Equations of Monotone Types in Banach Spaces , 2010 .

[12]  Aiki Toyohiko.  Multi-dimensional stefan problems with dynamic boundary conditions , 1995 .

[13]  J. Simon.  Compact sets in the spaceLp(O,T; B) , 1986 .

[14]  M. Brokate,et al.  Hysteresis and Phase Transitions , 1996 .

[15]  Jürgen Sprekels,et al.  Optimal Control of an Allen-Cahn Equation with Singular Potentials and Dynamic Boundary Condition , 2012, SIAM J. Control. Optim..

[16]  H. Brezis.  Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert , 1973 .

[17]  J. Cahn,et al.  A microscopic theory for antiphase boundary motion and its application to antiphase domain coasening , 1979 .

[18]  Alain Miranville,et al.  On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions , 2009 .

[19]  A. Miranville,et al.  The Penrose-Fife phase-field model with coupled dynamic boundary conditions , 2014 .

[20]  H. Israel,et al.  Long time behavior of an Allen-Cahn type equation with a singular potential and dynamic boundary conditions , 2012 .

[21]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[22]  Ciprian G. Gal,et al.  The non-isothermal Allen-Cahn equation with dynamic boundary conditions , 2008 .

[23]  P. Colli,et al.  Global solution to the Allen–Cahn equation with singular potentials and dynamic boundary conditions , 2012, 1206.6738.