- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 11:24:03
4
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=IFB&vol=5&iss=3&update_since=2024-03-28
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
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
5
2003
3
A geometrical approach to front propagation problems in bounded domains with Neumann-type boundary conditions
Guy
Barles
Université de Tours, TOURS, FRANCE
Francesca
Da Lio
Università di Torino, TORINO, ITALY
Front propagation, reaction-diffusion equations, asymptotic behavior, geometrical approach, level-set approach, Neumann boundary condition, angle boundary condition, viscosity solutions.
In this article, we are interested in the asymptotic behavior of the solutions of scaled reaction-diffusion equations, set in bounded domains, associated with Neumann type boundary conditions, and more precisely in cases when such behavior is described in terms of moving interfaces. A typical example is the case of the Allen-Cahn Equation associated with an oblique derivative boundary condition where the generation of a front moving by mean curvature with an angle boundary condition is shown. In order to rigourously establish such results, we modify and adapt the ``geometrical approach'' introduced by P.E.~Souganidis and the first author for solving problems set in $\R^N:$ we provide a new definition of weak solution for the global-in-time motion of fronts with curvature dependent velocities and with angle boundary conditions, which turns out to be equivalent to the level-set approach when there is no fattening phenomenon. We use this definition to obtain the asymptotic behavior of the solutions of a large class of reaction-diffusion equations, including the case of quasilinear ones and $(x,t)-$ dependent reaction terms, but also with any, possibly nonlinear, Neumann boundary conditions.
Partial differential equations
Numerical analysis
General
239
274
10.4171/IFB/79
http://www.ems-ph.org/doi/10.4171/IFB/79
A1-L1_0 phase boundaries and anisotropy via multiple-order-parameter theory for an fcc alloy
Gamze
Tanoglu
Izmir Institute of Technology, URLA, IZMIR, TURKEY
Richard
Braun
University of Delaware, NEWARK, UNITED STATES
John
Cahn
and Technology, GAITHERSBURG, UNITED STATES
Geoffrey
McFadden
and Technology, GAITHERSBURG, UNITED STATES
anisotropy; interphase boundaries; antiphase boundaries; fcc alloy
The dependence of thermodynamic properties of planar interphase boundaries (IPBs) and antiphase boundaries (APBs) in a binary alloy on an FCC lattice is studied as a function of their orientation. Using a recently-developed diffuse interface model based on three non-conserved order parameters and the concentration, and a free energy density that gives a realistic phase diagram with one disordered phase (A1) and two ordered phases (L$1_2$ and L$1_0$) such as occurs in the Cu-Au system,we are able to find IPBs and APBs between any pair of phases and domains, and for all orientations. The model includes bulk and gradient terms in a free energy functional, and assumes that there is no mismatch in the lattice parameters for the disordered and ordered phases. We catalog the appropriate boundary conditions for all IPBs and APBs. We then focus on the IPB between the disordered A1 phase and the L1$_0$ ordered phase. For this IPB we compute the numerical solution of the boundary value problem to find its interfacial energy, $\gamma$, as a function of orientation, temperature, and chemical potential (or composition). We determine the equilibrium shape for a precipitate of one phase within the other using the Cahn-Hoffman `$\xi$--vector' formalism. We find that the profile of the interface is determined only by one conserved and one non-conserved order parameter, which leads to a surface energy which, as a function of orientation, is ``transversely isotropic'' with respect to the tetragonal axis of the L1$_0$ phase. We verify the model's consistency with the Gibbs adsorption equation.
Partial differential equations
Numerical analysis
Mechanics of deformable solids
General
275
299
10.4171/IFB/80
http://www.ems-ph.org/doi/10.4171/IFB/80
A framework for the construction of level set methods for shape optimization and reconstruction
Martin
Burger
Johannes Kepler Universität Linz, LINZ, AUSTRIA
Level Sets, Shape Optimization, Shape Reconstruction, Inverse Problems, Hamilton-Jacobi Equations, Materials Science, Crystal Growth, Gradient Flows
The aim of this paper is to develop a functional-analytic framework for the construction of level set methods, when applied to shape optimization and shape reconstruction problems. As a main tool we use a notion of gradient flows for geometric configurations such as used in the modelling of geometric motions in materials science. The analogies to this field lead to a scale of level set evolutions, characterized by the norm used for the choice of the velocity. This scale of methods also includes the standard approach used in previous work on this subject as a special case. Moreover, we apply this framework to some (inverse) model problems for elliptic boundary value problems. In numerical experiments we demonstrate that an appropriate choice of norms (dependent on the problem) yields stable and fast methods.
Partial differential equations
Numerical analysis
Mechanics of deformable solids
General
301
329
10.4171/IFB/81
http://www.ems-ph.org/doi/10.4171/IFB/81
First variation of anisotropic energies and crystalline mean curvature of partitions
Giovanni
Bellettini
Università di Roma 'Tor Vergata', ROMA, ITALY
Matteo
Novaga
Università di Pisa, PISA, ITALY
Giuseppe
Riey
Università di Roma, ROMA, ITALY
crystalline mean curvature; anisotropic energy
We rigorously derive the notion of crystalline mean curvature af an anisotropic partition with no restriction on the space dimension. Our results cover the case of crystalline networks in two dimensions, polyhedral partitions in three dimensions, and generic anisotropic partitions for smooth anisotropies. The natural equilibrium conditions on the singular set of the partition are derived. We discuss several examples in two dimensions (also for two adjacent triple junctions)and one example in three dimensions when the Wulff shape is the unit cube. In the examples we analyze also the stability of the partitions.
Partial differential equations
Numerical analysis
Mechanics of deformable solids
General
331
356
10.4171/IFB/82
http://www.ems-ph.org/doi/10.4171/IFB/82