Home > Partial Differential Equations, shape optimization > Existence Result for the Isoperimetric Problems

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 BV(\Bbb{R}^N). By definition we have for an open set U \subset \Bbb{R}^N that BV(U)=\left\{ f \in L^1(U) : \sup \left\{\int_U f {\rm div} \varphi dx | \varphi \in C_c^1(U;\Bbb{R}^N),\ |\varphi|\leq 1\right\} \right\}.  Here we denoted by C_c^1(U;\Bbb{R}^N) the space of continuously differentiable functions f : U \to \Bbb{R}^N with compact support in U. Because of the density of the space C_c^\infty(U,\Bbb{R}^N) of infinitely differentiable functions f: U \to \Bbb{R}^N with compact support in U in the space C_c^1(U;\Bbb{R}^N), we could have replaced C_c^1 by C_c^\infty in the above definition. You could take a look at this blog post for a detailed description of BV(U) or at the Wikipedia page.

We say that a set A of finite Lebesgue measure is a set of finite perimeter in \Bbb{R}^N if its characteristic function \chi_A belongs to BV(\Bbb{R}^N). This means that the distributional gradient \nabla \chi_A is a vector valued measure with finite total variation. The total variation |\nabla \chi_A| is called the perimeter of A.

In the same way we can define the perimeter of a Lebesgue measurable set A relative to an open set D. We say that A\subset D is a set of finite perimeter relative to D if the characteristic function \chi_A belongs to the space BV(D).

The gradient \nabla \chi_A is be seen as a distribution due to the following relations:

\displaystyle \int_A {\rm div}\varphi dx= \int_D \chi_A \left( \sum_{i=1}^N \frac{\partial \varphi_i}{\partial x_i} \right) dx= =\langle \chi_A , \sum_{i=1}^N \frac{\partial \varphi_i}{\partial x_i}\rangle=\sum_{i=1}^N \langle \frac{\partial \chi_A}{\partial x_i},\varphi_i\rangle =-\langle \nabla \chi_A,\varphi \rangle

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 C^1. 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 D where the admissible domains belong is not bounded, then the shape optimization problem may have no solution, therefore we impose that the admissible domains A to be contained in a compact set K \subset \Bbb{R}^N. Instead of fixing the Lebesgue measure, we can impose the more general constraint \int_A f(x)dx=c where c is a given constant and f \in L^1_{loc} (\Bbb{R}^N). When f is the constant function 1 we get the usual volume constraint on A. The isoperimetric problem can be formulated as follows:

Given a compact subset K of \Bbb{R}^N and a function f \in L^1_{loc}(\Bbb{R}^N) find the subset of K whose perimeter is minimal among all subsets A of K for which \int_A f(x)dx=c, where c is given.

Denoting F(\Omega)=Per(\Omega)=\int |\nabla \chi_\Omega| and \mathcal{A}=\{ A \subset K : \int_A f(x)dx =c\} we have the following result:

Theorem: With the notations above, if K is a compact set and if \mathcal{A} is nonempty, then the minimization problem \min\{ F(A) : A \in \mathcal{A}\} admits at least a solution.

Proof: Since the class \mathcal{A} of admissible sets is not empty, there exists a minimizing sequence (A_n) with the property that F(A_n) \to \inf \{F(A) : A \in \mathcal{A}\}. Since A_n \subset K, it follows that the measures of A_n form a bounded sequence, and the perimeters Per(A_n) form also a bounded sequence. This means that \chi_{A_n} is a bounded sequence in BV(Q), where Q is a ball containing K. Then we can extract a subsequence (we denote it by the same A_n) which converges weakly* to some function u \in BV(Q) in the sense that

\begin{cases} \chi_{A_n} \to u \text{ strongly in }L^1(Q) \\ \nabla \chi_{A_n} \to \nabla u \text{ weakly * in the sense of measures} \end{cases}

The function u has to be of the form \chi_A for some set A with finite perimeter; moreover A \subset K (up to a set of measure zero) and \int_A f(x)fx=c, which shows that A \in \mathcal{A}. This domain achieves the minimum for the functional F, since the perimeter is weakly* lower semicontinuous function on BV(\Bbb{R}^N).

The same approach can be made to prove the existence in the case that F(\Omega)=Per_D(\Omega), the relative perimeter with respect to an open set D. 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.


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