33 research outputs found
Optimal control of an Allen-Cahn equation with singular potentials and dynamic boundary condition
In this paper, we investigate optimal control problems for Allen-Cahn
equations with singular nonlinearities and a dynamic boundary condition
involving singular nonlinearities and the Laplace-Beltrami operator. The
approach covers both the cases of distributed controls and of boundary
controls. The cost functional is of standard tracking type, and box constraints
for the controls are prescribed. Parabolic problems with nonlinear dynamic
boundary conditions involving the Laplace-Beltrami operation have recently
drawn increasing attention due to their importance in applications, while their
optimal control was apparently never studied before. In this paper, we first
extend known well-posedness and regularity results for the state equation and
then show the existence of optimal controls and that the control-to-state
mapping is twice continuously Fr\'echet differentiable between appropriate
function spaces. Based on these results, we establish the first-order necessary
optimality conditions in terms of a variational inequality and the adjoint
state equation, and we prove second-order sufficient optimality conditions.Comment: Key words: optimal control; parabolic problems; dynamic boundary
conditions; optimality condition
Optimal control of an Allen-Cahn equation with singular potentials and dynamic boundary condition
In this paper, we investigate optimal control problems for AllenCahn
equations with singular nonlinearities and a dynamic boundary condition
involving singular nonlinearities and the Laplace Beltrami operator. The
approach covers both the cases of distributed controls and of boundary
controls. The cost functional is of standard tracking type, and box
constraints for the controls are prescribed. Parabolic problems with
nonlinear dynamic boundary conditions involving the LaplaceBeltrami operation
have recently drawn increasing attention due to their importance in
applications, while their optimal control was apparently never studied
before. In this paper, we first extend known well-posedness and regularity
results for the state equation and then show the existence of optimal
controls and that the control-to-state mapping is twice continuously Fréchet
differentiable between appropriate function spaces. Based on these results,
we establish the firstorder necessary optimality conditions in terms of a
variational inequality and the adjoint state equation, and we prove
second-order sufficient optimality condition
Weak formulation for singular diffusion equation with dynamic boundary condition
In this paper, we propose a weak formulation of the singular diffusion
equation subject to the dynamic boundary condition. The weak formulation is
based on a reformulation method by an evolution equation including the
subdifferential of a governing convex energy. Under suitable assumptions, the
principal results of this study are stated in forms of Main Theorems A and B,
which are respectively to verify: the adequacy of the weak formulation; the
common property between the weak solutions and those in regular problems of
standard PDEs.Comment: 23 page
Global existence, uniqueness and stability for nonlinear dissipative bulk-interface interaction systems
We show global well-posedness and exponential stability of equilibria for a
general class of nonlinear dissipative bulk-interface systems. They correspond
to thermodynamically consistent gradient structure models of bulk-interface
interaction. The setting includes nonlinear slow and fast diffusion in the bulk
and nonlinear coupled diffusion on the interface. Additional driving mechanisms
can be included and non-smooth geometries and coefficients are admissible, to
some extent. An important application are volume-surface reaction-diffusion
systems with nonlinear coupled diffusion.Comment: 21 page
On the longtime behavior of a viscous Cahn-Hilliard system with convection and dynamic boundary conditions
In this paper, we study the longtime asymptotic behavior of a phase
separation process occurring in a three-dimensional domain containing a fluid
flow of given velocity. This process is modeled by a viscous convective
Cahn-Hilliard system, which consists of two nonlinearly coupled second-order
partial differential equations for the unknown quantities, the chemical
potential and an order parameter representing the scaled density of one of the
phases. In contrast to other contributions, in which zero Neumann boundary
conditions were are assumed for both the chemical potential and the order
parameter, we consider the case of dynamic boundary conditions, which model the
situation when another phase transition takes place on the boundary. The phase
transition processes in the bulk and on the boundary are driven by free
energies functionals that may be nondifferentiable and have derivatives only in
the sense of (possibly set-valued) subdifferentials. For the resulting
initial-boundary value system of Cahn-Hilliard type, general well-posedness
results have been established in a recent contribution by the same authors. In
the present paper, we investigate the asymptotic behavior of the solutions as
times approaches infinity. More precisely, we study the -limit (in a
suitable topology) of every solution trajectory. Under the assumptions that the
viscosity coefficients are strictly positive and that at least one of the
underlying free energies is differentiable, we prove that the -limit is
meaningful and that all of its elements are solutions to the corresponding
stationary system, where the component representing the chemical potential is a
constant.Comment: Key words: Cahn-Hilliard systems, convection, dynamic boundary
conditions, well-posedness, asymptotic behavior, omega-limit. arXiv admin
note: text overlap with arXiv:1704.0533
A boundary control problem for the pure Cahn-Hilliard equation with dynamic boundary conditions
A boundary control problem for the pure Cahn-Hilliard equations with possibly
singular potentials and dynamic boundary conditions is studied and first-order
necessary conditions for optimality are proved.
Key words: Cahn-Hilliard equation, dynamic boundary conditions, phase
separation, singular potentials, optimal control, optimality conditionsComment: arXiv admin note: text overlap with arXiv:1407.391
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 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.Comment: Key words: Allen-Cahn equation, dynamic boundary condition, mass
constraint, variational inequality, Lagrange multiplie
On the Cahn-Hilliard equation with dynamic boundary conditions and a dominating boundary potential
The Cahn-Hilliard and viscous Cahn-Hilliard equations with singular and
possibly nonsmooth potentials and dynamic boundary condition are considered and
some well-posedness and regularity results are proved. Key words: Cahn-Hilliard
equation, dynamic boundary conditions, phase separation, irregular potentials,
well-posedness.Comment: A revised version of this paper has been published on J. Math. Anal.
Appl. 419 (2014), 972-994: the authors would like to point out that both the
published version and the arXiv preprint contain a useless assumption in
formula (2.37), concerning the zero normal derivative of the initial value on
the boundary; indeed this condition is never employed in the proof
A boundary control problem for the viscous Cahn-Hilliard equation with dynamic boundary conditions
A boundary control problem for the viscous Cahn-Hilliard equations with
possibly singular potentials and dynamic boundary conditions is studied and
first order necessary conditions for optimality are proved.
Key words: Cahn-Hilliard equation, dynamic boundary conditions, phase
separation, singular potentials, optimal control, optimality conditions,
adjoint state syste