scholarly journals Half-space theorems for the Allen–Cahn equation and related problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
François Hamel ◽  
Yong Liu ◽  
Pieralberto Sicbaldi ◽  
Kelei Wang ◽  
Juncheng Wei
Keyword(s):  
Free Boundary ◽  
Half Space ◽  
Level Set ◽  
Free Boundary Problems ◽  
Entire Solution ◽  
Dimensional Solution ◽  
Boundary Problems ◽  
One Dimensional ◽  
Rigidity Results ◽  
Cahn Equation

AbstractIn this paper we obtain rigidity results for a non-constant entire solution u of the Allen–Cahn equation in {\mathbb{R}^{n}}, whose level set {\{u=0\}} is contained in a half-space. If {n\leq 3}, we prove that the solution must be one-dimensional. In dimension {n\geq 4}, we prove that either the solution is one-dimensional or stays below a one-dimensional solution and converges to it after suitable translations. Some generalizations to one phase free boundary problems are also obtained.

Heat Potentials Method in the Treatment of One-dimensional Free Boundry Problems Applied in Cryomedicine

2018 ◽  
Vol 85 (1-2) ◽  
pp. 111 ◽  
Author(s):  
Fatimat K. Kudayeva ◽  
Arslan A. Kaigermazov ◽  
Elizaveta K. Edgulova ◽  
Mariya M. Tkhabisimova ◽  
Aminat R. Bechelova
Keyword(s):  
Free Boundary ◽  
Integral Equations ◽  
Free Boundary Problems ◽  
Evolutionary Equation ◽  
Fredholm Integral Equations ◽  
Integral Expression ◽  
Boundary Problems ◽  
System Of Integral Equations ◽  
One Dimensional ◽  
Heat Potential

Free boundary problems are considered to be the most difcult and the least researched in the eld of mathematical physics. The present article is concerned with the research of the following issue: treatment of one-dimensional free boundary problems. The treated problem contains a nonlinear evolutionary equation, which occurs within the context of mathematical modeling of cryosurgery problems. In the course of the research, an integral expression has been obtained. The obtained integral expression presents a general solution to the non-homogeneous evolutionary equation which contains the functions that represent simple-layer and double-layer heat potential density. In order to determine the free boundary and the density of potential a system of nonlinear, the second kind of Fredholm integral equations was obtained within the framework of the given work. The treated problem has been reduced to the system of integral equations. In order to reduce the problem to the integral equation system, a method of heat potentials has been used. In the obtained system of integral equations instead of K(ξ; x; τ - t) in case of Dirichlet or Neumann conditions the corresponding Greens functions G(ξ; x; τ - t) or N(ξ; x; τ - t) have been applied. Herewith the integral expression contains fewer densities, but the selection of arbitrary functions is reserved. The article contains a number of results in terms of building a mathematical model of cooling and freezing processes of biological tissue, as well as their effective solution development.

A Four-Step Fixed-Grid Method for 1D Stefan Problems

2010 ◽  
Vol 132 (11) ◽  
Author(s):  
M. Tadi
Keyword(s):  
Free Boundary ◽  
Moving Boundary ◽  
Free Boundary Problems ◽  
Grid Method ◽  
Moving Boundary Problems ◽  
Boundary Problems ◽  
One Dimensional ◽  
Nonlinear Part ◽  
Stefan Problems ◽  
Fixed Grid

This note is concerned with a fixed-grid finite difference method for the solution of one-dimensional free boundary problems. The method solves for the field variables and the location of the boundary in separate steps. As a result of this decoupling, the nonlinear part of the algorithm involves only a scalar unknown, which is the location of the moving boundary. A number of examples are used to study the applicability of the method. The method is particularly useful for moving boundary problems with various conditions at the front.

Axisymmetric free boundary problems

1997 ◽  
Vol 341 ◽  
pp. 269-294 ◽  
Author(s):  
MARK SUSSMAN ◽  
PETER SMEREKA
Keyword(s):  
Free Surface ◽  
Free Boundary ◽  
Level Set ◽  
Free Boundary Problems ◽  
Immiscible Fluids ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Inviscid Limit ◽  
Navier Stokes Equation ◽  
Boundary Problems

We present a number of three-dimensional axisymmetric free boundary problems for two immiscible fluids, such as air and water. A level set method is used where the interface is the zero level set of a continuous function while the two fluids are solutions of the incompressible Navier–Stokes equation. We examine the rise and distortion of an initially spherical bubble into cap bubbles and toroidal bubbles. Steady solutions for gas bubbles rising in a liquid are computed, with favourable comparisons to experimental data. We also study the inviscid limit and compare our results with a boundary integral method. The problems of an air bubble bursting at a free surface and a liquid drop hitting a free surface are also computed.

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