Free Boundary problems and their applications

In this colloquial talk I shall present a few problems arising in various applied fields, that give rise to partial differential equations in a priori unknown domains; thus the term free boundary. These problems in general include phase transitions such: 

i)    the solid-liquid interphase when the temperature goes through the solidification of some material,

ii)   the boundary between exercise and continuation region for a financial instrument (such as option, and optimal stopping), 

iii)  the transition from elastic to plastic behavior when stress goes through a critical value,

iv)  the diffusion limited aggregation (controlled random walks on lattice and their continuous limits). 

I shall keep the talk at non-technical level, and will only explain the problems from various view points, stressing on the mathematical models and their formulations. A few results, tools and methodology will be mentioned.