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\TopicFR{\'Equations aux dérivées partielles, Théorie du contrôle}
\TopicEN{Partial differential equations, Control theory}

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\title{A formula for the sum of $n$ weak$^\star$ closed sets in $L^\infty$}

\author{\firstname{Mohsen} \lastname{Zivari-Rezapour}}
\address{Department of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Ahvaz, Iran}
\email{mzivari@scu.ac.ir}

\author{\firstname{Yichen} \lastname{Liu}\IsCorresp}
\address{Department of Applied Mathematics, School of Mathematics and Physics, Xi'an Jiaotong-Liverpool University, Suzhou, China}
\email{yichen.liu01@xjtlu.edu.cn}

\author{\firstname{Behrouz} \lastname{Emamizadeh}}
\address{Department of Mathematical Sciences, University of Nottingham Ningbo China, Ningbo, China}
\address{Institute for Research in Fundamental Sciences (IPM), Tehran, Iran}
\email{behrouz.emamizadeh@nottingham.edu.cn}
\email{behrouz.emamizadeh@ipm.ir}

\thanks{M. Zivari-Rezapour is grateful to the Research Council of Shahid Chamran University of Ahvaz for the research grant SCU.MM1402.441.
Y. Liu is supported by the Natural Science Foundation of Jiangsu Province (grant number: BK20200249)}
\CDRGrant[Research Council of Shahid Chamran University of Ahvaz]{SCU.MM1402.441}
\CDRGrant[Natural Science Foundation of Jiangsu Province]{BK20200249}

\begin{abstract}
In this note, we derive an equation which describes the closure of a particular set comprising $n-$valued functions. This result provides an answer to a long standing question for which the particular case $n=2$ had been known and used frequently in the optimal control problems.
\end{abstract}

\keywords{\kwd{$n-$valued functions}
\kwd{formulation}
\kwd{weak$^\star$ closure}
\kwd{rearrangements}
\kwd{eigenvalues}}
\subjclass{26E40, 49R05}


\begin{document}
\maketitle

\section{Introduction} In many optimal control problems, the admissible set is defined as follows
\begin{equation}\label{eqnd6}
A:=\left\{f\in L^\infty(D):f(f-1)=0,\,\int_{D}f\dx =\alpha\right\}
\end{equation}
where $D$ is a bounded (measurable) set in $\mathbb{R}^N$, $\alpha\in (0,|D|)$, and $|D|$ denotes the Lebesgue measure of $D$, see for example~\cite{HM01}. Henceforth, we identity $L^\infty$ as $\left(L^1\right)^\star$, the dual of $L^1$. Then, it is widely known that
\begin{equation}\label{eqnd1}
w^\star-\text{closure of }A=\left\{f\in L^\infty(D):0\le f\le 1,\,\int_Df\dx=\alpha\right\},
\end{equation}
see Proposition~2.4 in~\cite{CM90}. This latter set is drastically more convenient to work with compared to the former. For example, it is convex and weak$^\star$ compact in $L^\infty(D)$, whereas $A$ fails to retain either of these two properties.

In this note, we answer a long standing question; mainly, there is an analogy of~\eqref{eqnd1} for $n-$valued functions. More precisely, we replace the set of $\{0,1\}-$valued functions (verifying the integral constraint $\int_{D}f\dx=\alpha$), i.e. $A$, by an appropriate set of $n-$valued functions. A natural generalization of~\eqref{eqnd6} is the following set
\begin{equation*}
\tilde{A}=\left\{f\in L^\infty(D):(f-\gamma_1)(f-\gamma_2)\cdots(f-\gamma_n)=0,\,\int_{D}f\dx=\alpha\right\}.
\end{equation*}
However, a function $h\in\tilde{A}$ may not satisfy the condition $|\{x\in D:f(x)=\gamma_i\}|>0,$ $i=1,2,\dots,n,$ which we require, as illustrated by the following example:

\begin{exam*}
Let $D=(0,1)$, $\alpha=1$ and $\gamma_i=i-1$, $i=1,2,3$. Clearly, the set $\tilde{A}$ contains the following 2-valued and 3-valued controls:
\begin{equation*}
f_1=2\chi_{(\frac{1}{2},1)},\quad f_2=\chi_{(0,\frac{1}{3})}+2\chi_{(\frac{2}{3},1)}.
\end{equation*}
\end{exam*}

Note that this drawback is irrelevant to $A$ due to imposing the condition $\alpha\in (0,|D|)$. To overcome this obstacle, we first make the following observation that
\begin{equation*}
A=\left\{\,\chi_E:E\subseteq D,\,|E|=\alpha\right\}
\end{equation*}
where $\chi_E$ denotes the characteristic function of $E$. Furthermore, we make a second observation that
\begin{equation*}
\left\{\,\chi_E:E\subseteq D,\,|E|=\alpha\right\}=\mathcal{R}(\,\chi_F)
\end{equation*}
where $F$ can be any subset of $D$ verifying $|F|=\alpha$. Here, $\mathcal{R}(\,\chi_F)$ denotes the classical rearrangement class generated by $\chi_F$. In the following lines, we briefly recall the definition of two functions being rearrangements of each other.

For two (measurable) functions $f,g:D\to\mathbb{R}$, we say $f$ and $g$ are rearrangements of each other~if
\begin{equation*}
\lambda_f(\alpha)=\lambda_g(\alpha)\quad\text{for all }\alpha\in\mathbb{R},
\end{equation*}
where $\lambda_f$ (similarly $\lambda_g$) denotes the distribution function of $f$, i.e.
\[
\lambda_f(\alpha)=|\{x\in D:f(x)\ge\alpha\}|.
\]
We use the notation $\mathcal{R}(f)$ to indicate the set of functions defined on $D$ which are rearrangements of $f$; $\mathcal{R}(f)$ is called the rearrangement class generated by $f$. Note that if $f$ and $g$ are rearrangements of each other, they will generate the same class of rearrangements, i.e. $\mathcal{R}(f)=\mathcal{R}(g)$.

Now, for $c_i\in\mathbb{R}$, $i=1,2,\dots,n$, we introduce the set
\begin{equation*}
\mathcal{A}=\left\{\sum_{i=1}^{n}c_i\chi_{E_i}: E_i\subseteq D, \{E_i\}\;\text{are mutually disjoint}, |E_i|=\alpha_i\right\},
\end{equation*}
where $\alpha_i>0$ are prescribed satisfying $\sum_{i=1}^n\alpha_i= |D|$. Without loss of generality, we suppose $\{c_i\}$ is strictly increasing. Our main result is the following
\begin{theo*}
Using the notations mentioned above, the following equation holds
\begin{equation}\label{eqnd3}
\bar{\mathcal{A}}=\sum_{i=1}^{n}\mathcal{K}_i
\end{equation}
where\, $\bar{\mathcal{A}}$ denotes the weak$^\star$ closure of $\mathcal{A}$,
\begin{equation*}
\mathcal{K}_i=\left\{f\in L^\infty(D):0\le f\le c_i-c_{i-1},\,\int_Df\dx=(c_i-c_{i-1})\sum_{k=i}^{n}\alpha_k\right\}\quad\text{for all } i=2,\dots,n,
\end{equation*}
and $\mathcal{K}_1=\left\{c_1\right\}$. Here, $c_1$ denotes the constant function.
\end{theo*}

\begin{rema}
A direct consequence of the Theorem is the following useful observation. Suppose $\beta_i\in\mathbb{R}$ satisfies $0<\beta_n<\beta_{n-1}<\dots<\beta_1<|D|$ and
\begin{equation*}
\mathcal{K}_i=\left\{f\in L^\infty(D):0\le f\le 1,\,\int_Df\dx=\beta_i\right\}\quad\text{for all } i=1,\dots,n.
\end{equation*}
If we set $\beta_0=|D|$, $\beta_{n+1}=0$ and
\begin{equation*}
\mathcal{A}=\left\{\sum_{i=1}^{n+1}(i-1)\chi_{E_i}: E_i\subseteq D, \{E_i\}\;\text{are mutually disjoint}, |E_i|=\beta_{i-1}-\beta_i\right\},
\end{equation*}
then we have
\begin{equation*}
\bar{\mathcal{A}}=\sum_{i=1}^{n}\mathcal{K}_i.
\end{equation*}
\end{rema}

\section{Proof of the Theorem}

In the proof of the Theorem, we use the following important lemma from~\cite{AC22} which we include a sketch of its proof for the convenience of readers.

\begin{lemm*}
Let $f_1,f_2,\dots,f_n\in L^\infty(D)$ such that $\int_{D}f_if_j\dx=\int_{0}^{|D|}f_i^\vartriangle f_j^\vartriangle\ds$ for all $i,j=1,2,\dots,n$. Then,
\begin{equation*}
\sum_{i=1}^{n}\overline{\mathcal{R}(f_i)}=\overline{\mathcal{R}\left(\sum_{i=1}^{n}f_i\right)}.
\end{equation*}
Here, $f_i^\vartriangle$ denotes the decreasing rearrangement on $(0,|D|)$, i.e. $f_i^\vartriangle(s):=\max{\left\{\alpha:\lambda_{f_i}(\alpha)\ge s\right\}}$.
\end{lemm*}
\begin{proof}
We only sketch the proof here; the readers can refer to Theorem~3 in~\cite{AC22} for details.

The inclusion $\overline{\mathcal{R}\left(\sum_{i=1}^{n}f_i\right)}\subseteq\sum_{i=1}^{n}\overline{\mathcal{R}(f_i)}$ follows from Theorem~18.10 in~\cite{Day70}. Now, we show the inclusion in the other direction. To this end, we set $f=\sum_{i=1}^{n}f_i$ and utilize the condition that $\int_{D}f_if_j\dx=\int_{0}^{|D|}f_i^\vartriangle f_j^\vartriangle\ds$ to show the following
\begin{equation*}
0\le\int_{0}^{|D|}\left(f^\vartriangle-\sum_{i=1}^{n}f_i^\vartriangle\right)^2\ds\le
\int_{D}\left(f-\sum_{i=1}^{n}f_i\right)^2\dx=0.
\end{equation*}
This, in turn, implies
\begin{equation}\label{eqnd5}
\left(\sum_{i=1}^{n}f_i\right)^\vartriangle=\sum_{i=1}^{n}f_i^\vartriangle.
\end{equation}
Next, let $h=\sum_{i=1}^{n}h_i\in \sum_{i=1}^{n}\overline{\mathcal{R}(f_i)}$ with $h_i\in \overline{\mathcal{R}(f_i)}$ for $i=1,2,\dots,n$. We can use the standard rearrangement techniques and~\eqref{eqnd5} to derive
\begin{equation*}
\int_{0}^{|D|}h^\vartriangle \ds=\int_{0}^{|D|}\left(\sum_{i=1}^{n}f_i\right)^\vartriangle\ds
\quad\text{and}\quad
\int_{0}^{t}h^\vartriangle \ds\le\int_{0}^{t}\left(\sum_{i=1}^{n}f_i\right)^\vartriangle\ds,
\end{equation*}
for all $t\in (0,|D|)$. This implies $h\in \overline{\mathcal{R}\left(\sum_{i=1}^{n}f_i\right)}$, by~\cite{ATL89} or Proposition~5 in~\cite{AC22}. The proof of the Lemma is complete.
\end{proof}

\begin{proof}[Proof of the Theorem]
The proof of the Theorem is in fact an immediate consequence of the Lemma.

By setting $c_0=0$, the key to the proof will be the decomposition in~\eqref{eqnd2}. We define
\begin{equation*}
f_i=(c_i-c_{i-1})\chi_{\bigcup_{k=i}^nE_k},
\end{equation*}
$i=1,2,\dots,n$. We then set
\begin{equation}\label{eqnd2}
f=\sum_{i=1}^{n}f_i.
\end{equation}
Clearly, $\mathcal{A}=\mathcal{R}(f)$. On the other hand, for almost every $s\in(0,|D|)$, we have
\[
f_i^\vartriangle(s)=(c_{i}-c_{i-1})\chi_{(0,\sum_{k=i}^{n}\alpha_{k})}(s),
\]
$i=1,2,\ldots,n$. For $i,j\in\{1,2,\ldots,n\}$ with $i\leq j$, then we deduce, by direct computations, that
\begin{equation}\label{eqq}
\int_D f_i(x)f_j(x)\dx=(c_{i}-c_{i-1})(c_{j}-c_{j-1})\sum_{k=j}^{n}\alpha_{k}=\int_0^{|D|}f_i^\vartriangle(s)f_j^\vartriangle(s)\ds.
\end{equation}
From~\eqref{eqnd2}, \eqref{eqq} and the Lemma, we infer that
\[
\bar{\mathcal{A}}=\bar{\R(f)}=\bar{\R\left(\sum_{i=1}^n f_i\right)}=\sum_{i=1}^{n}\bar{\R(f_i)}.
\]
Moreover, by Proposition~2.4 in~\cite{CM90}, we have
\[
\mathcal{K}_i:=\bar{\R(f_i)}=\left\{g\in L^\infty(D):0\leq g\leq c_{i}-c_{i-1},\,\int_D g\dx=(c_{i}-c_{i-1})\sum_{k=i}^n\alpha_{k}\right\},
\]
$i=1,2,\dots,n$. Notice that $\R(f_1)=\{f_1\}=\{c_1\}$, hence we infer $\mathcal{K}_1=\bar{\R(f_{1})}=\left\{c_1\right\}$. Therefore, the decomposition~\eqref{eqnd3} follows.
\end{proof}

\begin{rema}
It is important to mention that everything stated in this paper holds true if the Lebesgue measure is replaced by any other (finite) non-atomic measures.
\end{rema}

This theorem has nice applications, for example, in spectral theory which we only outline here. Consider the eigenvalue problem
\begin{equation}\label{bvp1}
\begin{cases} 
-\Delta u=\lambda \rho(x)u, &\text{in } D\\
u=0, & \text{on } \partial D,
\end{cases}
\end{equation}
where this boundary value problem models the displacement, denoted $u$, of a clamped membrane with density $\rho(x)$ on the bounded domain $D$ from the rest position. It is well known that if $\rho$ is a non-trivial function in $L^\infty_+(D)$, then the eigenvalue problem~\eqref{bvp1} has a sequence of eigenvalues such that
\begin{equation*}
0<\lambda_1<\lambda_2\le\hdots\to\infty,
\end{equation*}
see for example~\cite{deF82}. $\lambda_1(\rho)$, to emphasize its dependence on $\rho$, is called the principal eigenvalue and it can be formulated in terms of the Rayleigh quotient
\begin{equation*}
\lambda_1(\rho):=\inf\left\{\frac{\int_{D}|\nabla v|^2\dx}{\int_{D}\rho v^2\dx}:v\in H_0^1(D),\,\int_{D}\rho v^2\dx>0\right\}.
\end{equation*}
One is interested in the following maximization and minimization problems
\begin{equation}\label{eqnd4}
\sup_{\rho\in \mathcal{K}_1+\mathcal{K}_2+\dots+\mathcal{K}_n}\lambda_1(\rho)
\quad\text{and}\quad
\inf_{\rho\in \mathcal{K}_1+\mathcal{K}_2+\dots+\mathcal{K}_n}\lambda_1(\rho).
\end{equation}
By using the Theorem, we can transform the above problems to the following rearrangement optimization problems
\begin{equation*}
\sup_{\rho\in \bar{\mathcal{A}}}\lambda_1(\rho)
\quad\text{and}\quad
\inf_{\rho\in \bar{\mathcal{A}}}\lambda_1(\rho)
\end{equation*}
which can be solved by the results in~\cite{CEP09,MPV13}. Moreover, one can prove that the optimization problems in~\eqref{eqnd4} have optimal solutions which, in turn, give rise to free boundary problems related to the eigenvalue problem~\eqref{bvp1}.

%\subsection*{Acknowledgement}
%M. Zivari-Rezapour is grateful to the Research Council of Shahid Chamran University of Ahvaz for the research grant [SCU.MM1402.441].
%Y. Liu is supported by the Natural Science Foundation of Jiangsu Province [grant number: BK20200249].



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