- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 15:33:01
5
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=IFB&vol=8&iss=4&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
8
2006
4
Optimal transportation networks as flat chains
Emanuele
Paolini
Università degli Studi di Firenze, FIRENZE, ITALY
Eugene
Stepanov
Università di Pisa, PISA, ITALY
We provide a model of optimization of transportation networks (e.g.\ urban traffic lines, subway or railway networks) in a geografical area (e.g. a city) with given density of population and that of services and/or workplaces, the latter being the destinations of everyday movements of the former. The model is formulated in terms of Federer-Fleming theory of currents, and allows to get both the position and the necessary capacity of the optimal network. Existence and some qualitative properties of solutions to the respective optimization problem are studied. Also, in an important particular case it is shown that the model proposed is equivalent to another known model of optimization of optimal transportation network, the latter not using the language of currents.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
393
436
10.4171/IFB/149
http://www.ems-ph.org/doi/10.4171/IFB/149
Uniqueness and least energy property for solutions to strongly competing systems
Monica
Conti
Politecnico, MILANO, ITALY
Susanna
Terracini
Università di Torino, TORINO, ITALY
Gianmaria
Verzini
Politecnico di Milano, MILANO, ITALY
For the reaction--diffusion system of three competing species: $$ -\D u_i=-\kappa u_i\sum_{j\neq i}u_j,\qquad i=1,2,3, $$ we prove uniqueness of the limiting configuration as $\kappa\to\infty$ on a planar domain $\O$, with appropriate boundary conditions. Moreover we prove that the limiting configuration minimizes the energy associated to the system $$ E(U)=\sum_{i=1}^3\int_\O|\nabla u_i(\bx)|^2\,d\bx $$ among all segregated states ($u_i\cdot u_j=0$ a.e.) with the same boundary conditions.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
437
446
10.4171/IFB/150
http://www.ems-ph.org/doi/10.4171/IFB/150
A degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions
Fuensanta
Andreu
Universitat de Valencia, BURJASSOT (VALENCIA), SPAIN
N.
Igbida
Université de Picardie Jules Verne, AMIENS CÉDEX 1, FRANCE
José
Mazón
Universitat de Valencia, BURJASSOT (VALENCIA), SPAIN
Julián
Toledo
Universitat de València, BURJASSOT (VALENCIA), SPAIN
In this paper we prove existence and uniqueness of weak solutions for a general degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions. Particular instances of this problem appear in various phenomena with changes of phase like multiphase Stefan problem and in the weak formulation of the mathematical model of the so called Hele Shaw problem. Also, the problem with non-homogeneous Neumann boundary condition is included.
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
447
479
10.4171/IFB/151
http://www.ems-ph.org/doi/10.4171/IFB/151
Crystalline curvature flow of planar networks
Giovanni
Bellettini
Università di Roma 'Tor Vergata', ROMA, ITALY
M.
Chermisi
Università di Roma 'Tor Vergata', ROMA, ITALY
Matteo
Novaga
Università di Pisa, PISA, ITALY
Motion by crystalline curvature of networks, polycrystalline materials
We consider the evolution of a polycrystalline material with three or more phases, in presence of an even crystalline anisotropy. We analyze existence, uniqueness, regularity and stability of the flow. In particular, if the flow becomes unstable at a finite time, we prove that an additional segment (or even an arc) at the triple junction may develop in order to decrease the energy and make the flow stable at subsequent times. We discuss some examples of collapsing situations that lead to changes of topology, such as the collision of two triple junctions.
Differential geometry
Mechanics of deformable solids
General
481
521
10.4171/IFB/152
http://www.ems-ph.org/doi/10.4171/IFB/152
Homogenization of contact line dynamics
Karl
Glasner
University of Arizona, TUCSON, UNITED STATES
This paper considers the effects of substrate inhomogeneity on the motion of the three phase contact line. The model employed assumes the slowness of the contact line in comparison to capillary relaxation. The homogenization of this free boundary problem with a spatially periodic velocity law is considered. Formal multiple scales analysis yields a local, periodic problem whose time-averaged dynamics correspond to the homogenized front velocity. A rigorous understanding of the long time dynamics is developed using comparison techniques. Computations employing boundary integral equations are used to illustrate the consequences of the analysis. Advancing and receding contact angles, pinning and anisotropic motion can be predicted within this framework.
Differential geometry
Mechanics of deformable solids
General
523
542
10.4171/IFB/153
http://www.ems-ph.org/doi/10.4171/IFB/153