## Relaxation of the Anisotropic Perimeter – Part 1

I have discussed in a previous post how Modica-Mortola theorem can provide a good framework for relaxing the perimeter functional in the single and multi-phase 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 Modica-Mortola 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 .

## Numerical Approximation using Relaxed Formulation

Sometimes it is easier to replace an optimization problem with a sequence of relaxed problems whose solutions approximate the solution to the initial problem.

This kind of procedure can be useful when we need to approximate numerically discontinuous functions (in particular the characteristic function). Modica Mortola theorem states that the functionals

-converges to the functional

(Recall that is a real function which is positive except for and where it is zero.)

## Master 6

(For the context see the Shape Optimization page where you can find links to the first 5 parts)

A particular consequence of the Modica-Mortola Theorem is that the functional

is lower semicontinuous with respect to the convergence for on the set

where the equalities are, as usual, up to a set of measure zero. It would be nice if a similar result would be true for multi-phase systems, where a functional of the form

is a -limit and therefore semicontinuous, for where

Let’s first remark that allowing the function in the Modica-Mortola theorem to have more than two zeros does not suffice. Indeed, if we allow to have zeros , then the limiting phase will take only two values and or and , depending on the constraint . This means that functionals of the form we presented above cannot be represented as a -limit when the function is scalar, but with more than two zeros. This obstacle can be overcome by passing to the multidimensional case. This approach is presented by Sisto Baldo in [1] and we will present the ideas of this approach below.

## Master 5

(If you are interested check out: Parts 1, 2, 3, 4, and the Shape Optimization page)

The notion of -convergence was introduced by E. De Giorgi and T. Franzioni in [1]. For an introduction in the subject see the books [2], [3], by Andrea Braides and the free document [4] by the same author.

Definition 1Let be a metric space, and for let be given . We say that -converges to on as , and we write or , if the following conditions hold:

- (LI) For every and every sequence such that in we have
- (LS) For every there exists a sequence such that in and

## Modica-Mortola Theorem

The notion of -convergence was introduced by E. De Giorgi and T. Franzioni in the article *Su un tipo di convergenza variazionale* 1975.

Let be a metric space, and for let be given . We say that -converges to on as , and we write , if the following conditions hold:

- (LI) For every and every sequence such that in we have
- (LS) For every there exists a sequence such that in and