## Shape Optimization course – Day 5

**Speaker: Edouard Oudet**

Consider the problem

where is a given functional and is a class of admissible sets.

One obvious way to represent a set is by using its characteristic function. In general, from a numerical point of view, it is harder to work with characteristic function than to work with smooth functions. For this we could think to associate to the initial problem a series of problems

where denotes here a smooth function which approximates the characteristic function in the sense that and therefore for small we have

If we try, for example, to approximate we may think of a formulation of the type

for , a smooth approximation of and a continuous, positive function with only two zeros, at and . Why could this be a valid choice? The integral of the gradient forces the minimizer to have a relatively small variation, and thus try and minimize the interface region, which in fact, gives the perimeter. The second term, the integral of , forces (when is small) the minimizer to be close to a characteristic function. Therefore we would expect that as the minimizers of should converge to minimizers of the perimeter.

This fact is made rigorous by using the -convergence. See more details in this post, where you can find a rigorous proof of the above approximation of the perimeter, called the Modica-Mortola theorem.

For the last two points of the course, I will point you to the webpage of Edouard Oudet, where more accurate details and pictures of the numerical results can be found.