Existence Result for the Isoperimetric Problems
The tricky part is how to define the perimeter of a Lebesgue measurable set with finite perimeter. This can be done considering the space of bounded variation functions, denoted . By definition we have for an open set that . Here we denoted by the space of continuously differentiable functions with compact support in . Because of the density of the space of infinitely differentiable functions with compact support in in the space , we could have replaced by in the above definition. You could take a look at this blog post for a detailed description of or at the Wikipedia page.
We say that a set of finite Lebesgue measure is a set of finite perimeter in if its characteristic function belongs to . This means that the distributional gradient is a vector valued measure with finite total variation. The total variation is called the perimeter of .
In the same way we can define the perimeter of a Lebesgue measurable set relative to an open set . We say that is a set of finite perimeter relative to if the characteristic function belongs to the space .
The gradient is be seen as a distribution due to the following relations:
This definition of the perimeter wouldn’t be of any good if it didn’t agree with the usual formula for the perimeter of bounded open sets of class . You can find a proof of this in this post following the lines from Henrot, Pierre, Variation et Optimization des Formes.
Having defined the perimeter of a Lebesgue measurable set in a general way, which agrees with the classical definition, we may now approach the proof of existence for the isoperimetric type problems presented below.
Let us first define a class of admissible domains for our problem. As it can be seen in this post, if the set where the admissible domains belong is not bounded, then the shape optimization problem may have no solution, therefore we impose that the admissible domains to be contained in a compact set . Instead of fixing the Lebesgue measure, we can impose the more general constraint where is a given constant and . When is the constant function we get the usual volume constraint on . The isoperimetric problem can be formulated as follows:
Given a compact subset of and a function find the subset of whose perimeter is minimal among all subsets of for which , where is given.
Denoting and we have the following result:
Theorem: With the notations above, if is a compact set and if is nonempty, then the minimization problem admits at least a solution.
Proof: Since the class of admissible sets is not empty, there exists a minimizing sequence with the property that . Since , it follows that the measures of form a bounded sequence, and the perimeters form also a bounded sequence. This means that is a bounded sequence in , where is a ball containing . Then we can extract a subsequence (we denote it by the same ) which converges weakly* to some function in the sense that
The function has to be of the form for some set with finite perimeter; moreover (up to a set of measure zero) and , which shows that . This domain achieves the minimum for the functional , since the perimeter is weakly* lower semicontinuous function on .
The same approach can be made to prove the existence in the case that , the relative perimeter with respect to an open set . This proves the existence result for the Dido problem presented here.
The above proof follows the lines from Dorin Bucur, Giuseppe Buttazzo, Variational Methods in Shape Optimization Problems.

November 5, 2011 at 7:08 pmNew definition for the perimeter of a set. « Problems – Beni Bogoşel

November 25, 2011 at 10:34 amShape Optimization Course – Day 1 « Problems – Beni Bogoşel

April 25, 2012 at 9:21 pmProof of the Isoperimetric Inequality « Problems – Beni Bogoşel

September 17, 2012 at 5:40 pmIntro to Shape Optimization « Problems – Beni Bogoşel

September 25, 2012 at 1:50 pmIntroduction to Shape Optimization « Free Boundaries