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.