Cahn-Hilliard models

Assume that we study a system compound of a material whose density u can take 2 values a and b (the two phases). This system is initially instable and we are intereseted in the solution of stability. The mass of the system is supposed to be constant.

initial instable system

The works conducted in the field of Cahn-Hilliard model type have shown that a solution of stability is given through the study of (with a constraint of constant mass) [1][4] :

This functional constitutes the two last term of the model we propose for image classification. W is the level constraint potential representing the Gibbs free energy.
The solution of the above problem, if it exits, has the following geometric properties :
u piecewise constant (almost everywhere), with exactly two admissible values : u(x)=a et u(x)=b.
the boundary between regions {x/u(x)=a} and {x/u(x)=b} is of minimal lenght.

a possible configuration of stability
This works have been extended to the case concerning n instable phases in [1]. You can find more details and references in the research report .