## Relaxation of the Anisotropic Perimeter – Part 2

(Read this previous post to see the context and the statement of the problem)

While it is usually difficult to construct a recovery sequence for an arbitrary (or all ), the construction is often much simpler if the target function has some special structure.

It is useful to consider a subset which is dense in such that for every there exists such that in and .

A common choice for is the class of characteristic functions of smooth sets, which is dense in in the sense that for every finite perimeter set there exists a sequence of smooth sets such that and as . Note that this is different than the BV-norm convergence, but it is the notion of convergence very well suited for BV functions. In Approximation of Free-Discontinuity Problems by A. Braides a proof of the limsup estimate is given using the dense class of smooth functions.

When dealing with the anisotropic perimeter, a more natural way of working would be choosing sets whose boundary is polyhedral (piecewise affine). The main reason for doing that is the fact that polyhedrons have constant normals corresponding to one face. To be able to choose as the class of characteristic functions of polyhedral subsets of (with finite perimeter, of course), we must have a density result similar to the smooth case.

## 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.)

## The Basic properties of Gamma Convergence

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

The -convergence has the following properties:

1. The -limit is always lower semicontinuous on ;

2. -convergence is stable under continuous perturbations: if and is continuous, then

3. If and minimizes over , then every limit point of minimizes over .

I have seen these properties stated in many places, but the proofs are usually left to the reader. I will try and give the proofs below.

## Master 7

(For the context and pervious posts look on the Shape Optimization page for the links)

As a result of the theorem proved in the previous post and of the fact that every -limit is lower semicontinuous we can see that the functional

is lower semicontinuous on the space .

Definition 1We say that a set of , is anadmissible configurationif there exist linearly independent vectors and a continuous function with zeros precisely at points such that we have .

Using the above result we can see that if is an admissible configuration then the functional defined by

is lower semicontinuous on

where is the unique solution of the system

where is chosen such that all are positive.

## 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.

## 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