- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 11:43:37
5
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=IFB&vol=3&iss=1&update_since=2024-03-29
Interfaces and Free Boundaries
Interfaces Free Bound.
IFB
1463-9963
1463-9971
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
10.4171/IFB
http://www.ems-ph.org/doi/10.4171/IFB
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
3
2001
1
On the Bernoulli free boundary problem and related shape optimization problems
Mohammed
Hayouni
Université Henri Poincaré, VANDOEUVRE LES NANCY, FRANCE
Antoine
Henrot
Université de Lorraine, VANDOEUVRE-LES-NANCY CEDEX, FRANCE
Nadia
Samouh
Université Moulay Ismaïl, BENI M'HAMED , MEKNÈS, MOROCCO
Bernoulli free boundary, continuous dependence, shape optimization
This paper deals with the classical Bernoulli free boundary problem. We are interested in solving some shape optimization problems related to this free boundary problem. We prove the continuous dependence of the solution with respect to the data K, working with Hausdorff convergence. We can deduce an existence result for a large class of shape optimization problems. Finally, we give some ideas for a numerical method, based on the use of conformal mappings, to solve such problems in two dimensions.
Functional analysis
Probability theory and stochastic processes
1
13
10.4171/IFB/30
http://www.ems-ph.org/doi/10.4171/IFB/30
A diffusion-convection problem with drainage arising in the ecology of mangroves
Cornelius
van Duijn
TU Eindhoven, EINDHOVEN, NETHERLANDS
Gonzalo
Galiano Casas
Universidad de Oviedo, OVIEDO, SPAIN
Mark
Peletier
Eindhoven University of Technology, EINDHOVEN, NETHERLANDS
Comparison principle, dead cores, existence, parabolic-elliptic PDEs, uniqueness
We consider both stationary and time-dependent versions of a model describing the vertical movement of water and salt in a porous medium in which a continuous extraction of water takes place (by the roots of mangroves). The problem is formulated in terms of a coupled system of partial differential equations for the salt concentration and the water flow which generalizes previous models. We study the existence and uniqueness of solutions and the conditions under which the maximum principle does hold, showing a counter-example for the general situation. We also analyse the stability of the steady state solution. Finally, we investigate the occurrence of dead cores (sets where the threshold salt concentration is attained) by means of the comparison principle in the stationary problem and of suitable energy estimates in the evolution problem.
Functional analysis
Probability theory and stochastic processes
15
44
10.4171/IFB/31
http://www.ems-ph.org/doi/10.4171/IFB/31
Three-phase boundary motion by surface diffusion: stability of a mirror symmetric stationary solution
KATSUO
ITO
KYUSHU UNIVERSITY, FUKUOKA, JAPAN
Yoshihito
Kohsaka
Faculty of Science, SAPPORO, JAPAN
We prove that the sharp interface model for a three-phase boundary motion by surface diffusion proposed by H. Garcke and A. Novick-Cohen admits a unique global solution provided the initial data fulfils a certain symmetric criterion and is also close to a minimizer of the energy under an area constraint. This minimizer is also a stationary solution of the present model. Moreover, we prove that the global solution converges to the minimizer of the energy as time goes to infinity.
Functional analysis
Probability theory and stochastic processes
45
80
10.4171/IFB/32
http://www.ems-ph.org/doi/10.4171/IFB/32
On the continuity of the free boundary in some class of two-dimensional problems
Michel
Chipot
Universität Zürich, ZÜRICH, SWITZERLAND
free boundary problems
We introduce a new method to show the continuity of the free boundary for problems in dimension two.
Functional analysis
Probability theory and stochastic processes
81
99
10.4171/IFB/33
http://www.ems-ph.org/doi/10.4171/IFB/33
The Cahn-Hilliard equation with elasticity-finite element approximation and qualitative studies
Harald
Garcke
Universität Regensburg, REGENSBURG, GERMANY
Martin
Rumpf
Universität Bonn, BONN, GERMANY
Ulrich
Weikard
Universität Bonn, BONN, GERMANY
We consider the Cahn-Hilliard equation-a fourth-order, nonlinear parabolic diffusion equation describing phase separation of a binary alloy which is quenched below a critical temperature. The occurrence of two phases is due to a nonconvex double well free energy. The evolution initially leads to a very fine microstructure of regions with different phases which tend to become coarser at later times. The resulting phases might have different elastic properties caused by a different lattice spacing. This effect is not reflected by the standard Cahn-Hilliard model. Here, we discuss an approach which contains anisotropic elastic stresses by coupling the expanded diffusion equation with a corresponding quasistationary linear elasticity problem for the displacements on the microstructure. Convergence and a discrete energy decay property are stated for a finite element discretization. An appropriate timestep scheme based on the strongly A-stable [Theta]-scheme and a spatial grid adaptation by refining and coarsening improve the algorithms efficiency significantly. Various numerical simulations outline different qualitative effects of the generalized model. Finally, a surprising stabilizing effect of the anisotropic elasticity is observed in the limit case of a vanishing fourth-order term, originally representing interfacial energy.
Functional analysis
Probability theory and stochastic processes
101
118
10.4171/IFB/34
http://www.ems-ph.org/doi/10.4171/IFB/34