- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 09:37:46
5
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=IFB&vol=4&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
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
4
2002
1
A codimension-two free boundary problem for the equilibrium shapes of a small three-dimensional island in an epitaxially strained solid film
L.
Shanahan
University at Buffalo, BUFFALO, UNITED STATES
Brian
Spencer
University at Buffalo, BUFFALO, UNITED STATES
Free boundary problem; elasticity; epitaxial film
We determine the equilibrium morphology of a strained solid film for the case where it wets the substrate (Stranski-Krastanow growth). Using a continuum elasticity model with isotropic surface energy and equal elastic constants in the film and substrate, we determine an asymptotic solution for the axisymmetric three-dimensional equilibrium shape of a small island, where the height is much less than the width, resulting in a codimension-two free boundary problem. This codimension-two free boundary problem can be reformulated as an integro-differential equation in which the island width appears as an eigenvalue. The solutions to the resulting integro-differential eigenvalue problem consist of a discrete spectrum of island widths and associated morphological modes, which are determined using a rapidly converging Bessel series. The lowest-order mode is energetically preferred and corresponds to the quantum dot morphology. Our predictions of quantum dot width compare favorably with experimental data in the GeSi/Si system. The higher-order modes, while not minimum-energy configurations, are similar to 'quantum ring` and 'quantum molecule` morphologies observed during the growth of strained films.
Functional analysis
Probability theory and stochastic processes
1
25
10.4171/IFB/50
http://www.ems-ph.org/doi/10.4171/IFB/50
Analysis of pricing American options on the maximum (minimum) of two risk assets
Lishang
Jiang
Tongji University, SHANGHAI, CHINA
American call/put options; parabolic obstacle problem; free boundary; penalized term
We use a PDE argument to deal with the mathematical analysis of the valuation of American options on the maximum/minimum of two assets. There are several factors which affect the valuation of options, such as stock price, strike price, the time to expiry, volatilities, the correlation constant, the risk-free interest rate and dividends. The first problem we are concerned with here is what happens to the prices of options if one of these factors is increasing while the others remain fixed. In the second part of this paper, the properties of the optimal exercise boundary of option as free boundary of the parabolic obstacle problem are studied such as monotonicity, convexity and asymptotic behavior.
Functional analysis
Probability theory and stochastic processes
27
46
10.4171/IFB/51
http://www.ems-ph.org/doi/10.4171/IFB/51
Existence and approximation of solutions to an anisotropic phase field system for the kinetics of phase transitions
Olaf
Klein
Angewandte Analysis und Stochastik, BERLIN, GERMANY
Phase-field model; anisotropy; semidiscretization; convergence
This paper is concerned with a phase field system of Penrose-Fife type for a non-conserved order parameter with a kinetic relaxation coefficient depending on the gradient of the order parameter. This system can be used to model the anisotropic solidification of liquids. A time-discrete scheme for an initial-boundary value problem to this system is presented. By proving the convergence of this scheme, the existence of a solution to the problem is shown.
Functional analysis
Probability theory and stochastic processes
47
70
10.4171/IFB/52
http://www.ems-ph.org/doi/10.4171/IFB/52
A total-variation surface energy model for thin films of martensitic crystals
Pavel
Belik
University of Minnesota, MINNEAPOLIS, UNITED STATES
Mitchell
Luskin
University of Minnesota, MINNEAPOLIS, UNITED STATES
Thin film; surface energy; bounded variation; martensite
We rigorously derive a thin-film limit for martensitic crystals that utilizes the total variation of the deformation gradient to model the energy on surfaces separating regions of different variants. We find that the deformation for an infinitesimally thin film minimizes a two-dimensional energy.
Functional analysis
Probability theory and stochastic processes
71
88
10.4171/IFB/53
http://www.ems-ph.org/doi/10.4171/IFB/53
A numerical scheme for axisymmetric solutions of curvature-driven free boundary problems, with applications to the Willmore flow
Uwe
Mayer
University of Utah, SALT LAKE CITY, UNITED STATES
Gieri
Simonett
Vanderbilt University, NASHVILLE, UNITED STATES
Willmore flow; numerical solutions; singularities
We present a numerical scheme for axisymmetric solutions to curvature-driven moving boundary problems governed by a local law of motion, e.g. the mean curvature flow, the surface diffusion flow, and the Willmore flow. We then present several numerical experiments for the Willmore flow. In particular, we provide numerical evidence that the Willmore flow can develop singularities in finite time.
Functional analysis
Probability theory and stochastic processes
89
109
10.4171/IFB/54
http://www.ems-ph.org/doi/10.4171/IFB/54