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.
Irrigation Problems.
Regularization of Images.
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