114 research outputs found
Equation and dynamic boundary condition of Cahn-Hilliard type with singular potentials
The well-posedness of a system of partial differential equations and dynamic
boundary conditions, both of Cahn-Hilliard type, is discussed. The existence of
a weak solution and its continuous dependence on the data are proved using a
suitable setting for the conservation of a total mass in the bulk plus the
boundary. A very general class of double-well like potentials is allowed.
Moreover, some further regularity is obtained to guarantee the strong solution
On a coupled bulk-surface Allen-Cahn system with an affine linear transmission condition and its approximation by a Robin boundary condition
We study a coupled bulk-surface Allen-Cahn system with an affine linear
transmission condition, that is, the trace values of the bulk variable and the
values of the surface variable are connected via an affine relation, and this
serves to generalize the usual dynamic boundary conditions. We tackle the
problem of well-posedness via a penalization method using Robin boundary
conditions. In particular, for the relaxation problem, the strong
well-posedness and long-time behavior of solutions can be shown for more
general and possibly nonlinear relations. New difficulties arise since the
surface variable is no longer the trace of the bulk variable, and uniform
estimates in the relaxation parameter are scarce. Nevertheless, weak
convergence to the original problem can be shown. Using the approach of Colli
and Fukao (Math. Models Appl. Sci. 2015), we show strong existence to the
original problem with affine linear relations, and derive an error estimate
between solutions to the relaxed and original problems.Comment: 34 page
Singular limit of Allen--Cahn equation with constraints and its Lagrange multiplier
We consider the Allen-Cahn equation with constraint. Our constraint is the subdifferential of the indicator function on the closed interval, which is the multivalued function. In this paper we give the characterization of the Lagrange multiplier to our equation. Moreover, we consider the singular limit of our system and clarify the limit of the solution and the Lagrange multiplier to our problem
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Singular limit of Allen-Cahn equation with constraints and its Lagrange multiplier
We consider the Allen-Cahn equation with constraint. Our constraint is
the subdifferential of the indicator function on the closed interval, which
is the multivalued function. In this paper we give the characterization of
the Lagrange multiplier to our equation. Moreover, we consider the singular
limit of our system and clarify the limit of the solution and the Lagrange
multiplier to our problem
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