Relaxation of the Anisotropic Perimeter – Part 1
I have discussed in a previous post how ModicaMortola theorem can provide a good framework for relaxing the perimeter functional in the single and multiphase cases. The ideas can be extended further to a more generalized notion of perimeter, the anisotropic perimeter. (anisotropic = directionally dependent)
The main idea is that the anisotropic perimeter doesn’t count every part of the boundary in the same way; some directions are more favorized than others. The anisotropic perimeter associated to a norm is defined by
There are variants of ModicaMortola theorem for the anisotropic perimeter. Here is one of them:
Theorem – Relaxation of the Anisotropic Perimeter
Let be a bounded open set with Lipschitz boundary. Let , let be a continuous function such that if and only if and let be a norm on . Let be defined by
and let be defined by
where . Then .
Proof: (the estimate) As usual, for the convergence proofs we have two parts. First we prove that for every as we have
Consider the function and let’s show that for with we have , where we use the notation
for every measure .
First note that if a.e then using the definition of the variation of a function we can see that . Moreover, if we have a function whose image contains only two real values then the absolutely continuous part and the Cantor part of are zero, while the jump part is
where is the normal to the jump set defined by . Having these in mind and using the fact that is homogeneous of degree one, we obtain
In order to prove the first property of the convergence we take two cases. First, assume that a.e. Then for in we have
If a.e. then applying Young’s inequality we get
where in the last inequality we have used Reshetnyak’s Theorem for the measures . This is one point where we use the convexity of the function . This finishes the first part of the proof.
This first part of the convergence proof can also be proved using a slicing technique and a reduction to the one dimensional case. For more details see Approximation of Free Discontinuity Problems by Andrea Braides.
I’ll come back with the estimate in a following post.

June 3, 2013 at 2:52 pmRelaxation of the Anisotropic Perimeter – Part 2  Beni Bogoşel's blog

September 20, 2015 at 11:37 pmOptimal partitioning of a square into rectangles  Beni Bogoşel's blog