scholarly journals Free boundary problems in biology

2015 ◽  
Vol 373 (2050) ◽  
pp. 20140368 ◽  
Author(s):  
Avner Friedman
Keyword(s):  
Mathematical Model ◽  
Wound Healing ◽  
Free Boundary ◽  
Existence And Uniqueness ◽  
Free Boundary Problems ◽  
Uniqueness Theorems ◽  
Biological Processes ◽  
Boundary Problems ◽  
Mathematical Problems ◽  
The Mathematical Model

In this paper, I review several free boundary problems that arise in the mathematical modelling of biological processes. The biological topics are quite diverse: cancer, wound healing, biofilms, granulomas and atherosclerosis. For each of these topics, I describe the biological background and the mathematical model, and then proceed to state mathematical results, including existence and uniqueness theorems, stability and asymptotic limits, and the behaviour of the free boundary. I also suggest, for each of the topics, open mathematical problems.

scholarly journals Mathematical modelling of Tyndall star initiation

2015 ◽  
Vol 26 (5) ◽  
pp. 615-645 ◽  
Author(s):  
A. A. LACEY ◽  
M. G. HENNESSY ◽  
P. HARVEY ◽  
R. F. KATZ
Keyword(s):  
Mathematical Model ◽  
Free Boundary ◽  
Free Boundary Problems ◽  
Liquid Interfaces ◽  
Boundary Problems ◽  
Volumetric Heating ◽  
Crystalline Anisotropy ◽  
Kinetic Undercooling ◽  
Interesting Class ◽  
Solid Liquid

The superheating that usually occurs when a solid is melted by volumetric heating can produce irregular solid–liquid interfaces. Such interfaces can be visualised in ice, where they are sometimes known as Tyndall stars. This paper describes some of the experimental observations of Tyndall stars and a mathematical model for the early stages of their evolution. The modelling is complicated by the strong crystalline anisotropy, which results in an anisotropic kinetic undercooling at the interface; it leads to an interesting class of free boundary problems that treat the melt region as infinitesimally thin.

Inverse free boundary problems for a generally degenerate parabolic equation

2015 ◽  
Vol 23 (2) ◽  
Author(s):  
Nadiya Huzyk
Keyword(s):  
Parabolic Equation ◽  
Free Boundary ◽  
Existence And Uniqueness ◽  
Degenerate Parabolic Equation ◽  
Free Boundary Problems ◽  
Time Dependent ◽  
Classical Solutions ◽  
Boundary Problems ◽  
First Derivative ◽  
Degenerate Parabolic

AbstractIn a free boundary domain we consider inverse problems for determination a time-dependent coefficient at the first derivative of an unknown function in a generally degenerate parabolic equation with different overdetermination conditions. There are established conditions of existence and uniqueness of the classical solutions to the named problems in the case of the weak degeneration.

scholarly journals On optimal regularity of free boundary problems and a conjecture of De Giorgi

2005 ◽  
Vol 58 (8) ◽  
pp. 1051-1076 ◽  
Author(s):  
Herbert Koch ◽  
Giovanni Leoni ◽  
Massimiliano Morini
Keyword(s):  
Free Boundary ◽  
Free Boundary Problems ◽  
Optimal Regularity ◽  
Boundary Problems
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