1. Introduction and main results
In this paper, we consider the weighted eigenvalue problem with homogenous Neumann boundary conditions
\begin{equation}
\begin{cases}-\Delta u =\lambda m u \quad &\text{in } \Omega\\
\cfrac{\partial u}{\partial \nu}=0 &\text{on } \partial\Omega, \end{cases}
\end{equation} where 
$\Omega\subset\mathbb{R}^N$ is an open bounded domain with Lipschitz boundary 
$\partial\Omega$, 
$m\in L^\infty(\Omega)$ changes sign in Ω, 
$\lambda\in\mathbb{R}$ and ν is the outward unit normal vector on 
$\partial \Omega$.
An eigenvalue λ of (1) is called principal eigenvalue if it admits a positive eigenfunction. Clearly, λ = 0 is a principal eigenvalue with positive constants as its eigenfunctions. Problem (1) has been studied in various papers (see, for example, [Reference Bocher8, Reference Brown and Lin9, Reference Senn and Hess44]). In particular, it is known that there is a positive (respectively negative) principal eigenvalue if and only if 
$\int_\Omega m\, dx \lt 0$ (respectively 
$\int_\Omega m\, dx \gt 0$). For the sake of completeness and in order to maintain this paper self-contained, we prefer to give here (see 
$\S$2) an independent proof of the result above. Moreover, we show that, under the previous hypothesis on m, there exists an increasing (respectively decreasing) sequence of positive (respectively negative) eigenvalues. The smallest positive eigenvalue is the principal eigenvalue, which will be denoted by 
$\lambda_1(m)$.
Problem (1) and its variants play a crucial role in studying nonlinear models from population dynamics (see [Reference Skellam45]) and population genetics (see [Reference Fleming26]). We illustrate in details the following model in population dynamics devised by Skellam [Reference Skellam45]
\begin{equation}
\begin{cases}
v_t=\Delta v+\gamma v[m(x)-v] \quad &\text{in }\Omega\times(0,\infty),\\
\cfrac{\partial v}{\partial \nu}\,=0 & \text{on } \partial\Omega\times(0,\infty),\\
v(x,0)=v_0(x)\geq 0, \ v(x,0)\not \equiv 0& \text{in } \overline{\Omega}.
\end{cases}
\end{equation} In (2), 
$v(x,t)$ represents the density of a population inhabiting the region Ω at location x and time t (for that reason, only non-negative solutions of (2) are of interest), v 0 is the initial density and γ is a positive parameter. The function m(x) represents the intrinsic local grow rate of the population, it is positive on favourable habitats and negative on unfavourable ones and it mathematically describes the available resources in the spatially heterogeneous environment Ω. The integral 
$\int_\Omega m\, dx$ can be interpreted as a measure of the total resources in Ω. The Neumann conditions in (2) are zero-flux boundary conditions: it means that no individuals cross the boundary of the habitat, i.e. the boundary acts as a barrier.
The model (2) can be also considered with Neumann boundary conditions replaced by the homogeneous Dirichlet or Robin conditions. In the first case, the environment Ω is surrounded by a completely inhospitable region, i.e. any individual reaching the boundary dies, while in the second some individuals reaching the boundary die and the others return to the interior of Ω. It is known (see [Reference Cantrell and Cosner13, Reference Cantrell and Cosner15] and references therein) that (2) predicts persistence for the population if 
$\lambda_1(m) \lt \gamma$. As a consequence, determining the best spatial arrangement of favourable and unfavourable habitats for the survival, within a fixed class of environmental configurations, results in minimizing 
$\lambda_1(m)$ over the corresponding class of weights. Having information of this type could affect, for example, on the strategies to be adopted for the conservation of species with limited resources.
This kind of problem has been investigated by many other authors. The question of determining the optimal spatial arrangements of favourable and unfavourable habitats in Ω for the survival of the modelled population was first addressed by Cantrell and Cosner in [Reference Cantrell and Cosner13, Reference Cantrell and Cosner14]. The authors considered the diffusive logistic equation (2) with homogeneous Dirichlet boundary conditions and when the weight m has fixed maximum, minimum and integral over Ω. The analogous problem with Neumann boundary conditions has been analysed by Lou and Yanagida in [Reference Lou and Yanagida36]. Berestycki et al. [Reference Berestycki, Hamel and Roques5] investigated a model similar to (2) in the case of periodically fragmented environment (
$\Omega=\mathbb{R}^N$ and m(x) periodic), Roques and Hamel [Reference Roques and Hamel42] proved the existence of a minimizer in the case of “bang-bang” configurations and also investigated the problem by using numerical computation, Jha and Porru [Reference Jha and Porru30], among other things, exhibited an example of symmetry breaking of the optimal arrangement of the local growth rate. Lamboley et al. [Reference Lamboley, Laurain, Nadin and Privat33] investigated model (2) with Robin boundary conditions. Mazari et al. [Reference Mazari, Nadin and Privat37] studied several shape optimization problems arising in population dynamics, we refer the reader to it for a review of current knowledge on the subject. We also mention Cadeddu et al. [Reference Cadeddu, Farina and Porru12], which considered mixed boundary conditions, Ferreri and Verzini [Reference Ferreri and Verzini25] which studied asymptotic properties for Dirichlet boundary conditions, Mazzoleni et al. [Reference Mazzoleni, Pellacci and Verzini38] which considered a singular analysis for Neumann problems, Derlet et al. [Reference Derlet and Gossez22] and Cuccu et al. [Reference Cuccu and Emamizadeh19], that extended these type of results to the principal eigenvalue associated to the p-Laplacian operator for Neumann and Dirichlet boundary conditions respectively. Finally, Pellacci and Verzini [Reference Pellacci and Verzini40] considered the fractional Laplacian operator and Dipierro et al. [Reference Dipierro, Proietti Lippi and Valdinoci23] a mixed local and nonlocal operator.
In order to present our work, we briefly give some notations and definitions here. We denote by 
$\lambda_k(m)$, 
$k\in \mathbb{N}$, the kth positive eigenvalue of problem (1) corresponding to the weight m (assuming 
$ \int_\Omega m\,dx \lt 0$). We say that two Lebesgue measurable functions 
$f,g:\Omega \to \mathbb{R}$ are equimeasurable if the superlevel sets 
$\{x\in \Omega: f(x) \gt t\}$ and 
$\{x\in \Omega: g(x) \gt t\}$ have the same measure for all 
$t\in \mathbb{R}$. For a fixed 
$f\in L^\infty(\Omega)$, we call the set 
$\mathcal G(f)=\{g:\Omega\rightarrow\mathbb{R}:g\ \text{is measurable and }g\;\textit{and}\ f\ \text{are equimeasurable}\}$ the class of rearrangements of f (see Appendix A). Moreover, we introduce the set 
$L^\infty_ \lt (\Omega)=\left\{m\in L^\infty (\Omega): \int_\Omega m\,dx \lt 0\right\}$.
The present paper contains three main results. First, we study the dependence of 
$\lambda_k(m)$ on m, in particular we investigate continuity and, for k = 1, convexity and differentiability properties (see Lemmas 3, 4 and 5). Second, we examine the optimization of 
$\lambda_1(m)$ in the class of rearrangements 
$\mathcal{G}(m_0)$ of a fixed function 
$m_0\in L^\infty_ \lt (\Omega)$. Precisely, we prove the existence of minimizers, a characterization of them in terms of the eigenfunctions relative to 
$\lambda_1(m)$ and a non-existence result for the maximizers.
Theorem 1. Let 
$\lambda_1(m)$ be the principal eigenvalue of problem (1), 
$m_0\in L_ \lt ^\infty(\Omega)$ such that the set 
$\{x\in \Omega:m_0(x) \gt 0\}$ has positive Lebesgue measure, 
$\mathcal{G}(m_0)$ the class of rearrangements of m 0 (see Definition 7) and 
$\overline{\mathcal{G}(m_0)}$ its weak* closure in 
$L^\infty(\Omega)$. Then
(i) the problem
\begin{equation}
\min_{m\in{\overline{\mathcal{G}(m_0)}}\atop |\{m \gt 0\}| \gt 0} \lambda_1(m)
\end{equation} admits solutions and any solution 
$\check{m}_1$ belongs to 
$\mathcal{G}(m_0)$;
(ii) for every solution 
$\check{m}_1\in\mathcal{G}(m_0)$ of (3), there exists an increasing function ψ such that
\begin{equation}
\check{m}_1= \psi(u_{\check{m}_1}) \quad \text{a.e. in }\Omega,
\end{equation} where 
$u_{\check{m}_1}$ is the unique positive eigenfunction relative to 
$\lambda_1(\check{m}_1)$ normalized as in (28);
(iii)
\begin{equation*}
\sup_{m\in{\mathcal{G}(m_0)}} \lambda_1(m)=+\infty.
\end{equation*} We note that the class of weights usually considered in literature, i.e. a set of bounded functions with fixed maximum, minimum and integral over Ω, can be written in terms of 
$\overline{\mathcal{G}(m_0)}$ for a m 0 which takes exactly two values (functions of this kind are called of “bang-bang” type). This fact is proved in [Reference Anedda and Cuccu2].
From the biological point of view, (i) of Theorem 1 says that there exists an arrangement of the resources that maximizes the chances of survival and, in this case, the population density is larger where the habitat is more favourable. On the other hand, (iii) means that there are configurations of resources as bad (i.e. inhospitable) as one prescribes.
Our third main result is the following
Theorem 2. Let 
$\Omega=(0,h)\times\omega\subset\mathbb{R}^N$, h > 0 and 
$\omega\subset\mathbb{R}^{N-1}$ be a bounded polyhedral or smooth domain. Let 
${m}_0\in L_ \lt ^\infty(\Omega)$ such that the set 
$\{x\in \Omega:m_0(x) \gt 0\}$ has positive Lebesgue measure and 
$\mathcal{G}(m_0)$ the class of rearrangements of m 0 (see Definition 7). Then every minimizer of (3) is monotone with respect to x 1, where x 1 is the first coordinate of 
$\mathbb{R}^N$.
Monotonicity results of this kind have been studied both theoretically and numerically by a number of authors. Theorem 2 in the one dimensional case has been proved in [Reference Cantrell and Cosner14, Reference Lou and Yanagida36] in the case m 0 is a “bang-bang” function and in [Reference Jha and Porru30] for general m 0. In general dimension, when the domain Ω is an orthotope and m 0 is of “bang-bang” type, Lamboley et al. in [Reference Lamboley, Laurain, Nadin and Privat33] show that any minimizer is monotonic with respect to every coordinate direction. Theorem 2 contains all previous results and it is coherent with numerical simulations in [Reference Kao, Lou and Yanagida31, Reference Roques and Hamel42] for rectangles and “bang-bang” weights.
It is worth mentioning that in the case (1) is considered with Dirichlet boundary conditions, the monotonicity of minimizers is replaced by the Steiner symmetry of them (see [Reference Anedda and Cuccu2, Reference Biswas, Das and Ghosh7, Reference Cantrell and Cosner14]). Nevertheless, in both situations these qualitative properties of the minimizers lead to an arrangement of the favourable resources fragmented as little as possible. Indeed, in the Dirichlet case they are concentrated far from the boundary, while in the Neumann case they meet the boundary.
As final remark, we observe that problem (1), with Dirichlet boundary conditions in place of Neumann and in the case of positive weight m(x), also has a well-known physical interpretation: it models the vibration of a membrane Ω with clamped boundary 
$\partial\Omega$ and mass density m(x); 
$\lambda_1(m)$ represents the principal natural frequency of the membrane. Therefore, physically, minimizing 
$\lambda_1(m)$ means to find the mass distribution of the membrane which gives the lowest principal natural frequency. Usually, the composite membrane is built using only two homogeneous materials of different densities and, then, the weights in the optimization problem take only two positive values. Among many papers that consider the optimization of the principal natural frequency, we recall [Reference Chanillo, Grieser, Imai, Kurata and Ohnishi16–Reference Cox and McLaughlin18].
This paper is structured as follows. In 
$\S$2, we set up the functional framework and some tools in order to investigate the spectrum of problem (1). In 
$\S$3, we study the dependence of 
$\lambda_k(m)$ on m, in particular continuity and, for k = 1, convexity and differentiability properties; then, we prove Theorem 1. In 
$\S$4, we give the proof of Theorem 2. Finally, in Appendix A we collect some known results about rearrangements of measurable functions we need to examine the optimization problem (3).
2. Notations, preliminaries and weak formulation of (1)
Let 
$\Omega \subset \mathbb{R}^N$, 
$N\geq 1$, be a bounded connected open set with Lipschitz boundary 
$\partial \Omega$.
In this paper, we denote by 
$|E|$ the measure of an arbitrary Lebesgue measurable set 
$E \subset \mathbb{R}^N$ and by 
$L^\infty(\Omega)$, 
$L^2(\Omega)$ and 
$H^1(\Omega)$ the usual Lebesgue and Sobolev spaces. The usual norms and scalar products of these spaces are denote by
\begin{equation*}
\|u\|_{L^\infty(\Omega)}={\mathrm{ess}\sup\,}_{\Omega} |u| \quad \forall\, u\in L^\infty(\Omega),
\end{equation*}
\begin{equation*}
\langle u,v \rangle_{L^2(\Omega)}=\int_\Omega uv \, dx \quad \forall\, u, v\in L^2(\Omega)
\end{equation*}
\begin{equation*}
\|u\|_{L^2(\Omega)}=\langle u,u \rangle^{1/2}_{L^2(\Omega)} \quad \forall\, u\in L^2(\Omega),
\end{equation*}
\begin{equation*}
\langle u,v \rangle_{H^1(\Omega)}=\int_{\Omega}uv\, dx +\int_\Omega \nabla u \cdot \nabla v \, dx
\quad \forall\, u, v\in H^1(\Omega),
\end{equation*}
\begin{equation}
\|u\|_{H^1(\Omega)}=\langle u,u \rangle_{H^1(\Omega)}^{1/2} \quad \forall\, u\in H^1(\Omega).
\end{equation} Moreover, we also use the notation 
$\langle \nabla u,\nabla v \rangle_{L^2(\Omega)}=\int_\Omega \nabla u\cdot \!\nabla v \, dx$ for all 
$u, v\in H^1(\Omega)$ and by weak* convergence we always mean the weak* convergence in 
$L^\infty(\Omega)$.
Given 
$m\in L^2(\Omega)$ such that m ≠ 0, we define the spaces
\begin{equation*}L^2_m(\Omega) = \left\{f\in L^2(\Omega): \int_\Omega m f\, dx =0\right\}\quad
\text{and}\quad V_m(\Omega) = H^1(\Omega)\cap L^2_m(\Omega).\end{equation*} 
$L^2_m(\Omega)$ and 
$V_m(\Omega)$ are separable Hilbert subspaces of 
$L^2(\Omega)$ and 
$H^1(\Omega)$ respectively.
2.1. The projection Pm and norm in 
$V_m(\Omega)$
In this subsection, we introduce a fundamental tool in order to develop our theory: a projection from 
$L^2(\Omega)$ to 
$L^2_m(\Omega)$ (which must not be confused with the usual orthogonal projection in Hilbert spaces).
Definition 1. Let 
$m\in L^2(\Omega)$ such that 
$\int_\Omega m\, dx \neq 0$. We call 
$\it{projection}$ Pm the operator
\begin{equation*}
P_m: L^2(\Omega) \to L^2_m(\Omega),\qquad f\mapsto f- \frac{\int_\Omega mf \, dx}{\int_\Omega m\, dx}\,.
\end{equation*} Note that 
$P_m(H^1(\Omega))\subset V_m(\Omega)$. Indeed, depending on the case (which will be clear from the context), it might be more convenient to consider the projection 
$P_m: H^1(\Omega) \to V_m(\Omega)$. Since in our work m(x) represents the local growth rate, which is a bounded function, hereafter we consider 
$m\in L^\infty(\Omega)$. Nevertheless, Proposition 1, Proposition 2 and Proposition 3 can also be stated for 
$m\in L^2(\Omega)$.
Proposition 1. Let 
$m,q\in L^\infty(\Omega)$ such that 
$\int_\Omega m\, dx, \int_\Omega q\, dx \neq 0$ and Pm the projection of Definition 1. Then
i)
$\langle mP_m(f), \varphi\rangle_{L^2(\Omega)}= \langle m f, P_m(\varphi)\rangle_{L^2(\Omega)}$ for all
$f,\varphi\in L^2(\Omega)$;ii)
$P_m(f)=0$ if and only if f is constant;iii)
$P_m(f)=f$ for all
$f\in L_m^2(\Omega)$;iv)
$\nabla P_m(f)=\nabla f$ for all
$f\in H^1(\Omega)$;v) Pm is a linear bounded operator with
(6)\begin{equation}
\|P_m\|_{\mathcal{L}(L^2(\Omega),L^2(\Omega)) }\leq 1+ \frac{\|m\|_{L^\infty(\Omega)}}{\left|\int_\Omega m\, dx \right|}\, |\Omega|
\end{equation}and
(7)\begin{equation}
\|P_m\|_{\mathcal{L}(H^1(\Omega),H^1(\Omega))}\leq 1+ \frac{\|m\|_{L^\infty(\Omega)}}{\left|
\int_\Omega m\, dx \right|}\, |\Omega|;
\end{equation}vi) the compositions
$P_q\circ P_m: L^2_q(\Omega) \to L^2_q(\Omega)$,
$P_q\circ P_m:
V_q(\Omega) \to V_q(\Omega)$ are identities;vii)
$ P_m: L^2_q(\Omega) \to L^2_m(\Omega)$,
$P_m: V_q(\Omega) \to V_m(\Omega)$ are isomorphisms.
Proof. (i), (ii), (iii) and (iv) are immediate consequences of the definition of the projection Pm.
(v) By the definition of Pm and straightforward calculations we find
\begin{equation*}
\begin{aligned}
\|P_m(f)\|^2_{L^2(\Omega)} &=
\int_\Omega\left[f^2-2\,\frac{\int_\Omega mf\;dx}{\int_\Omega m\;dx}f+\left(\frac{\int_\Omega mf\;dx}{\int_\Omega m\;dx}\right)^2 \right]\;dx\\
&= \|f\|_{L^2(\Omega)}^2-2\,\frac{\int_\Omega mf\;dx}{\int_\Omega m\;dx}\int_\Omega f\;dx+\left(\frac{\int_\Omega mf\;dx}{\int_\Omega m\;dx}\right)^2|\Omega|\\
&\leq \|f\|_{L^2(\Omega)}^2+2\,\frac{\|m\|_{L^2(\Omega)}\|f\|^2_{L^2(\Omega)}}{\left|\int_\Omega m\;dx
\right|}|\Omega|^{1/2}+\,\frac{\|m\|^2_{L^2(\Omega)}\|f\|^2_{L^2(\Omega)}}{\left|\int_\Omega m\;dx
\right|^2}|\Omega|\\
&=\left(\|f\|_{L^2(\Omega)}+\,\frac{\|m\|_{L^2(\Omega)}\|f\|_{L^2(\Omega)}}{\left|\int_\Omega m\;dx\right|}|\Omega|^{1/2}\right)^2\\
&\leq \left(\|f\|_{L^2(\Omega)}+\,\frac{\|m\|_{L^\infty(\Omega)}\|f\|_{L^2(\Omega)}}{\left|\int_\Omega m\;dx\right|}|\Omega|\right)^2\\
& = \left(1+\,\frac{\|m\|_{L^\infty(\Omega)}}{\left|\int_\Omega m\;dx\right|}|\Omega|\right)^2\|f\|^2_{L^2(\Omega)};\\
\end{aligned}
\end{equation*}then (6) holds. The estimate (7) follows from (6) and (iv).
(vi) Let 
$f\in L^2_q(\Omega)$; recalling that 
$\int_\Omega qf\, dx = 0$, we have
\begin{align*}P_q (P_m(f)) &= P_q \left(f- \frac{\int_\Omega mf \, dx}{\int_\Omega m\, dx} \right)\\
&= f- \frac{\int_\Omega mf \, dx}{\int_\Omega m\, dx}\, -
\frac{\int_\Omega qf\,dx- \frac{\int_\Omega mf \, dx}{\int_\Omega m\, dx}\int_{\Omega}q\, dx}
{\int_\Omega q\,dx} =f;\end{align*}the second statement immediately follows from the first one.
(vii) It immediately follows from vi).
For the sake of convenience, we put
\begin{equation}
C_1(m)=1+ \frac{\|m\|_{L^\infty(\Omega)}}{\left|\int_\Omega m\, dx \right|}\, |\Omega|.
\end{equation} The previous proposition leads us to an alternative norm in the space 
$V_m(\Omega)$.
Proposition 2. Let 
$m\in L^\infty(\Omega)$ such that 
$\int_\Omega m\, dx\neq 0$. Then, for all 
$u\in V_m(\Omega)$ we have
\begin{equation}
\|u\|_{L^2(\Omega)}\leq C\left(1+ \frac{\|m\|_{L^\infty(\Omega)}}{\left|\int_\Omega m\, dx \right|}\,
|\Omega|\right)\|\nabla u\|_{L^2(\Omega)},
\end{equation}with C 2 equal to the constant of the Poincaré-Wirtinger’s inequality (see [Reference Leoni35, Theorem 12.23]).
Proof. Let 
$u\in V_m(\Omega)$. By (vi) of Proposition 1 we have 
$u=P_m(P_1(u))$. By (6) and the Poincaré-Wirtinger’s inequality we find
\begin{equation*}
\begin{aligned}
\|u\|_{L^2(\Omega)}=\|P_m(P_1(u))\|_{L^2(\Omega)}&\leq C_1(m)\|P_1(u)\|_{L^2(\Omega)}\\
&\leq C_1(m)\left \|u - \frac{1}{|\Omega|} \int_\Omega u\, dx\right\|_{L^2(\Omega)}\\
&\leq C_1(m)\cdot C \|\nabla u \|_{L^2(\Omega)},
\end{aligned}
\end{equation*}which proves the statement.
Proposition 3. Let 
$m\in L^\infty(\Omega)$ such that 
$\int_\Omega m\, dx\neq 0$. Then, the bilinear form in 
$V_m(\Omega)$
\begin{equation} \langle u,v \rangle_{V_m(\Omega)}=\int_\Omega \nabla u \cdot \nabla v\, dx \quad \forall\, u, v\in V_m(\Omega)
\end{equation} is a scalar product which induces a norm equivalent to the usual norm (5). We denote by 
$\|u\|_{V_m(\Omega)}$ the associated norm to (10).
Proof. Comparing 
$\|u\|_{V_m(\Omega)}$ with (5), we have
\begin{equation}
\|u\|_{V_m(\Omega)}\leq \|u\|_{H^1(\Omega)}.
\end{equation}
\begin{equation}
\|u\|_{H^1(\Omega)}\leq (C^2\cdot C_1^2(m)+1)^{1/2}\|u\|_{V_m(\Omega)}.
\end{equation}If not stated otherwise, we will consider Vm endowed with the norm just introduced.
2.2. The operators Em and Gm
We study the eigenvalues of problem (1) by means of the spectrum of an operator that we will introduce in this subsection.
Let 
$m\in L^\infty(\Omega)$ such that 
$\int_\Omega m\, dx\neq 0$. For every 
$f\in L^2(\Omega)$ let us consider the following continuous linear functional on 
$V_m(\Omega)$
\begin{equation*}
\varphi\mapsto
\langle m f,\varphi\rangle_{L^2(\Omega)} \quad\forall\, \varphi\in V_m(\Omega).
\end{equation*} By the Riesz Theorem, there exists a unique 
$u\in V_m(\Omega)$ such that
\begin{equation}
\langle u, \varphi\rangle_{V_m(\Omega)}
= \langle m f,\varphi\rangle_{L^2(\Omega)} \quad\forall\, \varphi\in V_m(\Omega)
\end{equation}holds.
Let us introduce the operator
\begin{equation}
E_m:L^2(\Omega)\to V_m(\Omega),
\end{equation} where 
$u=E_m(f)$ is the unique function in 
$V_m(\Omega)$ that satisfies (13), i.e. for all 
$f\in L^2(\Omega)$, 
$E_m(f)$ is defined by
\begin{equation}
\langle E_m(f) , \varphi\rangle_{V_m(\Omega)}
= \langle m f,\varphi\rangle_{L^2(\Omega)} \quad\forall\, \varphi\in V_m(\Omega).
\end{equation} Em is clearly linear. Putting 
$\varphi=u$ in (13) and exploiting (9) and (8), we find
\begin{equation}
\|u\|_{V_m(\Omega)}\leq C\cdot C_1(m)\|m\|_{L^\infty(\Omega)}\|f\|_{L^2(\Omega)}.
\end{equation}Therefore Em is a linear bounded operator such that
\begin{equation}
\|E_m\|_{\mathcal{L}(L^2(\Omega), V_m(\Omega))}\leq C\cdot C_1(m) \|m\|_{L^\infty(\Omega)}.
\end{equation} Let im be the inclusion of 
$V_m(\Omega)$ into 
$L^2(\Omega)$. Note that, by compactness of the inclusion 
$H^1(\Omega) \hookrightarrow L^2(\Omega) $ (see [Reference Leoni35]), it follows that im is a compact operator as well. Moreover, we define a second the linear operator
\begin{equation}
G_m:V_m(\Omega)\to V_m(\Omega)
\end{equation} by 
$G_m=E_m\circ i_m$, i.e. for all 
$f\in V_m(\Omega)$, 
$G_m(f)$ is defined by
\begin{equation}
\langle G_m(f) , \varphi\rangle_{V_m(\Omega)}
= \langle m f,\varphi\rangle_{L^2(\Omega)} \quad\forall\, \varphi\in V_m(\Omega).
\end{equation}The main properties of the operators Em and Gm are summarized in the following Proposition.
Proposition 4. Let 
$m\in L^\infty(\Omega)$ such that 
$\int_\Omega m\, dx\neq 0$ and 
$P_m, E_m$ and Gm defined by Definition 1, (15) and (19) respectively. Then
i)
$E_m(f)= G_m( P_m(f))$ for all
$f\in H^1(\Omega)$;ii) Gm is self-adjoint and compact;
iii) Em restricted to
$H^1(\Omega)$ is compact.
Proof. (i) Let 
$f\in H^1(\Omega)$, then 
$P_m(f)\in V_m(\Omega)$. By (19), (i) and (iii) of Proposition 1 we have
\begin{align*}
\langle G_m(P_m(f)), \varphi\rangle_{V_m(\Omega)}
&= \langle mP_m(f), \varphi
\rangle_{L^2(\Omega)}=
\langle mf, P_m(\varphi)\rangle_{L^2(\Omega)}\\&=\langle mf, \varphi\rangle_{L^2(\Omega)}\quad\forall\,\varphi\in
V_m(\Omega).
\end{align*} Thus, by (15), 
$E_m(f)= G_m( P_m(f))$.
(ii) For all 
$f, g\in V_m(\Omega)$, by (19), we have
\begin{equation*}
\langle G_m(f), g\rangle_{V_m(\Omega)} =
\langle m f, g\rangle_{L^2(\Omega)}
= \langle m g, f\rangle_{L^2(\Omega)}
=\langle G_m (g), f\rangle_{V_m(\Omega)},
\end{equation*} then Gm is self-adjoint. The compactness of the operator Gm is an immediate consequence of its definition 
$G_m=E_m \circ i_m $, the inclusion im being compact and the operator Em continuous.
(iii) It follows from (i) and (ii).
By the general theory of self-adjoint compact operators (see [Reference de Figueiredo, Dold and Eckmann21, Reference Lax34]), it follows that all nonzero eigenvalues of Gm have a finite dimensional eigenspace and they can be obtained by the Fischer’s Principle
\begin{align} \mu_k(m) &= \sup_{F_k\subset V_m(\Omega)}
\inf_{f\in F_k\atop f\neq 0}
\cfrac{\langle G_m (f), f \rangle_{V_m(\Omega)} }{\|f\|^2_{V_m(\Omega)}} \,\\&= \sup_{F_k\subset V_m(\Omega)}
\inf_{f\in F_k\atop f\neq 0}
\cfrac{\int_\Omega m f^2 \, dx }{\int_\Omega |\nabla f|^2\, dx} \,, \quad k=1, 2, 3, \ldots
\end{align}and
\begin{align*}
\mu_{-k}(m) &= \inf_{F_k\subset V_m(\Omega)}
\sup_{f\in F_k\atop f\neq 0}
\cfrac{\langle G_m (f), f \rangle_{V_m(\Omega)} }{\|f\|^2_{V_m(\Omega)}} \,\\&=
\inf_{F_k\subset V_m(\Omega)}
\sup_{f\in F_k\atop f\neq 0}
\cfrac{\int_\Omega m f^2 \, dx }{\|f\|^2_{V_m(\Omega)}}\,,\quad k=1, 2, 3, \ldots,
\end{align*} where the first extrema are taken over all the subspaces Fk of 
$V_m(\Omega)$ of dimension k. As observed in [Reference de Figueiredo, Dold and Eckmann21], all the inf’s and sup’s in the above characterizations of the eigenvalues are actually assumed. Hence, they could be replaced by min’s and max’s and the eigenvalues are obtained exactly in correspondence of the associated eigenfunctions. The sequence 
$\{\mu_k(m)\}$ contains all the real positive eigenvalues (repeated with their multiplicity), is decreasing and converging to zero, whereas 
$\{\mu_{-k}(m)\}$ is formed by all the real negative eigenvalues (repeated with their multiplicity), is increasing and converging to zero.
We will write 
$\{m \gt 0\}$ as a short form of 
$\{x\in \Omega: m(x) \gt 0\}$ and similarly 
$\{m \lt 0\}$ for 
$\{x\in \Omega: m(x) \lt 0\}$. The following proposition is analogous to [Reference de Figueiredo, Dold and Eckmann21, Proposition 1.11].
Proposition 5. Let 
$m \in L^\infty(\Omega)$ and Gm be the operator (19). Then, the following statements hold
i) if
$|\{m \gt 0\}|=0$, then there are no positive eigenvalues;ii) if
$|\{m \gt 0\}| \gt 0$ and
$\int_\Omega m\, dx \lt 0$, then there is a sequence of positive eigenvalues
$\mu_k(m)$;iii) if
$|\{m \lt 0\}|=0$, then there are no negative eigenvalues;iv) if
$|\{m \lt 0\}| \gt 0$ and
$\int_\Omega m\, dx \gt 0$, then there is a sequence of negative eigenvalues
$\mu_{-k}(m)$.
Proof. (i) Let µ be an eigenvalue and u a corresponding eigenfunction. By (19) with 
$f=\varphi=u$ we have
\begin{equation*}\mu= \cfrac{\int_\Omega m u^2\, dx }{\|u\|^2_{V_m(\Omega)}}\, \leq 0.\end{equation*} (ii) By measure theory covering theorems, for each positive integer k there exist k disjoint closed balls 
$B_1, \ldots, B_k$ in Ω such that 
$| B_i \cap \{m \gt 0\}| \gt 0$ for 
$i=1, \ldots, k$. Let 
$f_i\in C^\infty_0(B_i)$ such that 
$\int_\Omega m f_i^2 \, dx=1$ for every 
$i=1, \ldots, k$. Note that the functions fi are linearly independent. We put 
$g_i=P_m(f_i)$, where Pm is defined in Definition 1; 
$g_i\in V_m(\Omega)$ for all 
$i=1, \ldots, k$. We show that the functions gi are linearly independent as well. Let 
$\alpha_1, \ldots , \alpha_k$ be constants such that 
$\sum_{i=1}^k \alpha_i g_i=0$; this implies 
$P_m\left(\sum_{i=1}^k \alpha_i f_i\right)=0$, i.e., by (ii) of Proposition 1, 
$\sum_{i=1}^k \alpha_i f_i=c\in\mathbb{R}$. Evaluating 
$\sum_{i=1}^k \alpha_i f_i$ in 
$\Omega\smallsetminus \cup_{i=1}^k B_i\neq
\emptyset$ we find c = 0 and, therefore, 
$\alpha_i=0$ for all 
$i=1, \ldots, k$.
Let 
$F_k= \ $span
$ \{g_1, \ldots, g_k\}$. Fk is a subspace of 
$V_m(\Omega)$ of dimension k. For every 
$g\in F_k\smallsetminus
\{0\}$, 
$g=\sum_{i=1}^k a_i g_i$, with suitable constants 
$a_i\in \mathbb{R}$. Let us put 
$f=\sum_{i=1}^k a_i f_i$, clearly 
$g=P_m(f)$. Then, by (i) and (iii) of Proposition 1 and recalling that 
$\int_\Omega m\, dx \lt 0$, we have
\begin{equation*} \begin{aligned}\cfrac{\langle G_m (g),
g\rangle_{V_m(\Omega)}}{\|g\|^2_{V_m(\Omega)}}\, = &
\cfrac{\langle mg, g\rangle_{L^2(\Omega)}}{\|g\|^2_{V_m(\Omega)}} \, =
\cfrac{\langle mP_m(f), P_m(f)\rangle_{L^2(\Omega)}}{\|g\|^2_{V_m(\Omega)}} \, =
\cfrac{\langle mf, P_m(f)\rangle_{L^2(\Omega)}}{\|g\|^2_{V_m(\Omega)}} \, \\=&
\cfrac{\langle mf, f\rangle_{L^2(\Omega)}-\left\langle mf, \int_\Omega mf\,dx /
\int_\Omega m\,dx\right\rangle_{L^2(\Omega)}}{\|g\|
^2_{V_m(\Omega)}} \, \\=&
\cfrac{\langle mf, f\rangle_{L^2(\Omega)}-\left(\int_\Omega mf\,dx\right)^2/
\int_\Omega m\,dx}{\|g\|^2_{V_m(\Omega)}} \,\\ \geq &
\cfrac{\sum_{i, j=1}^k a_i a_j\int_\Omega m f_i f_j \,dx}{\sum_{i,j=1}^k\langle g_i, g_j\rangle_{V_m(\Omega)} a_i a_j} \,
=\cfrac{\sum_{i=1}^k a_i^2}{\sum_{i,j=1}^k\langle g_i, g_j\rangle_{V_m(\Omega)} a_i a_j} \,\\&=
\cfrac{\|a\|^2_{\mathbb{R}^k}}{\langle A_k a, a\rangle_{\mathbb{R}^k}}\,\geq \cfrac{1}{\|A_k\|}\, \gt 0,
\end{aligned}
\end{equation*} where 
$\|a\|_{\mathbb{R}^k}$, 
$\|A_k\|$ and 
$\langle A_k a, a\rangle_{\mathbb{R}^k}$ denote, respectively, the euclidean norm of the vector 
$a=(a_1, \ldots, a_k)$, the norm of the non null matrix 
$A_k=\left( \langle g_i, g_j\rangle_{V_m(\Omega)} \right)_{i, j=1}^k$ and the inner product in 
$\mathbb{R}^k$. From the Fischer’s Principle (20) we conclude that 
$\mu_k(m)\geq \cfrac{1}{\|A_k\|}\, \gt 0$ for every k.
The cases (iii) and (iv) are similarly proved.
2.3. Weak formulation of problem (1)
The operators Em and Gm are related to the following problem with Neumann boundary conditions
\begin{equation}
\begin{cases}-\Delta u = mf \quad &\text{in } \Omega\\
\cfrac{\partial u}{\partial \nu}=0 &\text{on } \partial \Omega.
\end{cases}
\end{equation} For 
$m\in L^{\infty}(\Omega)$ and 
$f\in L^2(\Omega)$, a weak solution of problem (21) is a function 
$u\in H^1(\Omega)$ such that
\begin{equation*}
\int_\Omega \nabla u\cdot \nabla\varphi \, dx = \int_\Omega m f\varphi\, dx \quad\forall\, \varphi\in H^1(\Omega)
\end{equation*}or, equivalently,
\begin{equation}
\left\langle \nabla u,\nabla\varphi\right\rangle_{L^2(\Omega)}
=\langle m f,\varphi\rangle_{L^2(\Omega)} \quad\forall\, \varphi\in H^1(\Omega).
\end{equation} The assumptions under which problem (21) admits solutions are well known (see for example [Reference Mikhailov39]). By using the tools introduced in 
$\S$2.2, we find those conditions independently.
Lemma 1. Let 
$m \in L^\infty(\Omega)$ such that 
$\int_\Omega m\, dx \neq 0$ and 
$f\in L^2(\Omega)$. Then 
$u\in H^1(\Omega)$ satisfies
\begin{equation}
\left\langle \nabla u,\nabla\varphi\right\rangle_{L^2(\Omega)}
=\langle m P_m(f),\varphi\rangle_{L^2(\Omega)} \quad\forall\, \varphi\in H^1(\Omega)
\end{equation} if and only if 
$P_m(u)$ satisfies
\begin{equation}
\left\langle P_m(u),\varphi\right\rangle_{V_m(\Omega)}
=\langle m f,\varphi\rangle_{L^2(\Omega)} \quad\forall\, \varphi\in V_m(\Omega).
\end{equation}Proof. If 
$u\in H^1(\Omega)$ satisfies (23), for all 
$\varphi\in V_m(\Omega)$, by (iv), (i) and (iii) of Proposition 1, we have
\begin{align*}
\left\langle P_m(u),\varphi\right\rangle_{V_m(\Omega)}
&=\left\langle\nabla P_m(u),\nabla\varphi\right\rangle_{L^2(\Omega)}
=\left\langle\nabla u,\nabla\varphi\right\rangle_{L^2(\Omega)}
\\&=\langle m P_m(f),\varphi\rangle_{L^2(\Omega)}=\langle mf,\varphi\rangle_{L^2(\Omega)}.
\end{align*} Vice versa, let u verify (24). For all 
$\varphi\in H^1(\Omega)$, recalling (iv) and (i) of Proposition 1 we have
\begin{equation*}
\begin{aligned}
\left\langle \nabla u,\nabla\varphi\right\rangle_{L^2(\Omega)}
&=\left\langle\nabla P_m(u),\nabla P_m(\varphi)\right\rangle_{L^2(\Omega)}
=\left\langle P_m(u), P_m(\varphi)\right\rangle_{V_m(\Omega)}\\
&=\langle m f,P_m(\varphi)\rangle_{L^2(\Omega)}
=\langle m P_m(f),\varphi\rangle_{L^2(\Omega)}.
\end{aligned}
\end{equation*}As the following proposition says, the operator Em provides the solutions of problem (21).
Proposition 6. Let 
$m \in L^\infty(\Omega)$ such that 
$\int_\Omega m\, dx \neq 0$ and Em be the operator (14). Then
i) (22) has a solution if and only if
$f\in L^2_m(\Omega)$;ii) if
$f\in L^2_m(\Omega)$, (22) has a unique solution
$\overline{u}\in
V_m(\Omega)$ and any other solution is of form
$\overline{u} + c$,
$c\in\mathbb{R}$;iii)
$\overline{u}=E_m(f)$, i.e.
$\overline{u}$ is the unique solution of
(iv) the estimate
\begin{equation*}\langle \overline{u},\varphi\rangle_{V_m(\Omega)}=\langle m f,\varphi\rangle_{L^2(\Omega)}
\quad\forall\varphi\in V_m(\Omega);\end{equation*}
\begin{equation*}
\|\overline{u}\|_{H^1(\Omega)}\leq C\cdot C_1(m)\left(C^2 \cdot C_1^2(m)+1\right)^{1/2} \|m\|_{L^\infty(\Omega)}\|f\|_{L^2(\Omega)}
\end{equation*}holds.
Proof. If (22) admits a solution u, choosing 
$\varphi\equiv 1$ we obtain 
$f\in L^2_m(\Omega)$.
Vice versa, let 
$f\in L^2_m(\Omega)$. By Lemma 1, 
$u\in H^1(\Omega)$ is a solution of (22) if and only if 
$P_m(u)$ is a solution of (24). By (13) and (15), we know that (24) admits the unique solution 
$\overline{u}=E_m(f)\in V_m(\Omega)$. Then the set of solutions of (22) is 
$\{u\in H^1(\Omega):P_m(u)=\overline{u}\}=\{u\in H^1(\Omega):u=\overline{u}+c,\, c\in\mathbb{R}\}$ and only 
$\overline{u}$ belongs to 
$V_m(\Omega)$. This proves (i), (ii) and (iii).
Finally, we introduce the weak formulation of problem (1). A function 
$u\in H^1(\Omega)$ is said an eigenfunction of (1) associated to the eigenvalue λ if
\begin{equation*}
\int_\Omega \nabla u\cdot \nabla\varphi \, dx = \lambda \int_\Omega m u\varphi\, dx \quad\forall\, \varphi\in H^1(\Omega)
\end{equation*}or, equivalently,
\begin{equation}
\langle \nabla u,\nabla \varphi\rangle_{L^2(\Omega)}
=\lambda \langle m u,\varphi\rangle_{L^2(\Omega)} \quad\forall\, \varphi\in H^1(\Omega).
\end{equation}It is easy to check that zero is an eigenvalue and the associated eigenfunctions are all of the constant functions.
Proposition 7. Let 
$m \in L^\infty(\Omega)$ such that 
$\int_\Omega m\, dx \neq 0$ and Gm be the operator (18). Then the nonzero eigenvalues of problem (1) are exactly the reciprocals of the nonzero eigenvalues of the operator Gm and the correspondent eigenspaces coincide.
Proof. If λ ≠ 0 is an eigenvalue and u is an associated eigenfunction of problem (1), choosing 
$\varphi\equiv 1$ in (25), we obtain 
$u\in V_m(\Omega)$. By (25) and (10) we have
\begin{equation*}
\left\langle \frac{u}{\lambda},\varphi\right\rangle_{V_m(\Omega)}
=\langle m u,\varphi\rangle_{L^2(\Omega)} \quad\forall\, \varphi\in V_m(\Omega)
\end{equation*} and then, by definition (19) of Gm, 
$G_m (u)= \cfrac{u}{\lambda}\, $.
Vice versa, let 
$G_m(u)=\mu u$, with µ ≠ 0. Then we have
\begin{equation*}
\left\langle \mu u,\varphi\right\rangle_{V_m(\Omega)}
=\langle m u,\varphi\rangle_{L^2(\Omega)} \quad\forall\, \varphi\in V_m(\Omega).
\end{equation*}By (iii) of Proposition 1 we obtain
\begin{equation*}
\left\langle P_m(\mu u),\varphi\right\rangle_{V_m(\Omega)}
=\langle m u,\varphi\rangle_{L^2(\Omega)} \quad\forall\, \varphi\in V_m(\Omega),
\end{equation*}using Lemma 1 we find
\begin{equation*}
\mu\left\langle\nabla u,\nabla\varphi\right\rangle_{L^2(\Omega)}
=\langle m P_m(u),\varphi\rangle_{L^2(\Omega)} \quad\forall\, \varphi\in H^1(\Omega)
\end{equation*}and finally, applying (iii) of Proposition 1 again, we conclude
\begin{equation*}
\left\langle\nabla u,\nabla\varphi\right\rangle_{L^2(\Omega)}
=\frac{1}{\mu}\langle m u,\varphi\rangle_{L^2(\Omega)} \quad\forall\, \varphi\in H^1(\Omega),
\end{equation*} i.e. 
$1/\mu$ is an eigenvalue of (1).
Consequently, in general, the eigenvalues of problem (1) form two monotone sequences
\begin{equation*} 0 \lt \lambda_1(m)\leq \lambda_2(m)\leq\ldots\leq \lambda_k(m)\leq \ldots\end{equation*}and
\begin{equation*} \ldots\leq\lambda_{-k}(m)\leq\ldots\leq \lambda_{-2}(m)\leq \lambda_{-1}(m) \lt 0 ,\end{equation*}where every eigenvalue appears as many times as its multiplicity, the latter being finite owing to the compactness of Gm.
The variational characterization (20) for k = 1, assuming that 
$|\{m \gt 0\}| \gt 0$ and 
$\int_\Omega m\,dx \lt 0$, becomes
\begin{equation} \mu_1(m) = \max_{f\in V_m(\Omega)\atop f\neq 0}
\cfrac{\langle G_m (f), f \rangle_{V_m(\Omega)} }{\|f\|^2_{V_m(\Omega)}} \,=\max_{f\in V_m(\Omega)\atop f\neq 0}
\cfrac{\int_\Omega m f^2 \, dx }{\int_\Omega |\nabla f|^2\, dx}.
\end{equation} The maximum in (26) is obtained if and only if f is an eigenfunction relative to µ 1. Similarly, for 
$\lambda_1(m)$ we have
\begin{equation}\lambda_1(m) = \min_{u\in V_m(\Omega)\atop \int_\Omega m u^2dx \gt 0}
\cfrac{\int_\Omega |\nabla u|^2\, dx}{\int_\Omega m u^2 \, dx}\,\end{equation}and the minimum in (27) is obtained if and only if u is an eigenfunction relative to λ 1.
We note that the characterization
\begin{equation*}
\quad\lambda_1(m) = \min_{u\in H^1(\Omega)\atop \int_\Omega m u^2dx \gt 0}
\cfrac{\int_\Omega |\nabla u|^2\, dx}{\int_\Omega m u^2 \, dx}\,
\end{equation*}also holds and it is more often used in the literature.
Proposition 8. Let 
$m\in L^\infty(\Omega)$ such that 
$|\{m \gt 0\}| \gt 0$ and 
$\int_\Omega m\, dx \lt 0$. Then 
$\mu_1(m)$ is simple and any associated eigenfunction is one signed in Ω.
Proof. Let 
$u\in V_m(\Omega)$ be an eigenfunction related to 
$\mu_1(m)$. Let us show that 
$|u|$ is an eigenfunction as well. Consider the projection 
$P_m(|u|)$ of 
$|u|$ on 
$V_m(\Omega)$, where Pm is defined in Definition 1. By (26) and (iv) of Proposition 1 we have
\begin{equation*}
\mu_1(m)\geq \cfrac{\int_\Omega m (P_m(|u|))^2 \, dx }{\int_\Omega |\nabla P_m(|u|)|^2\, dx}=
\cfrac{\int_\Omega m u^2 \, dx-\frac{\left(\int_\Omega m |u|\, dx\right)^2}{\int_\Omega m\,dx}}{\int_\Omega |\nabla u|^2\, dx}
\geq\cfrac{\int_\Omega m u^2 \, dx }{\int_\Omega |\nabla u|^2\, dx}=\mu_1(m).
\end{equation*} Therefore, we have the equality sign in the previous chain. In particular, we find 
$\int_\Omega m |u|
\, dx=0$, then 
$|u|$ belongs to 
$V_m(\Omega)$ and finally, by (iii) of Proposition 1, 
$|u|
=P_m(|u|)$ is an eigenfunction. By Proposition 7, 
$|u|$ satisfies the equation 
$-\Delta |u|
=\mu_1(m)^{-1} m|u|$ and, by Harnack inequality (see [Reference Gilbarg and Trudinger27]), we conclude that 
$|u| \gt 0$ in Ω; therefore u is one signed in Ω. Let 
$u,v$ be two eigenfunctions; set 
$\alpha=\frac{\int_\Omega v\,dx}{\int_\Omega u\,dx}$, note that 
$\int_\Omega (\alpha u- v)\, dx=0$. Note that also 
$\alpha u-v$ is an eigenfunction of 
$\mu_1(m)$. If 
$\alpha u-v$ was not identically zero, then, it would be one signed and hence 
$\int_\Omega (\alpha u- v)\, dx\neq0$, reaching a contradiction. Therefore 
$v=\alpha u$ and 
$\mu_1(m) $ is simple.
As a consequence of Proposition 7, we have the following
Corollary 1. Let 
$m\in L^\infty(\Omega)$ such that 
$|\{m \gt 0\}| \gt 0$ and 
$\int_\Omega m\, dx \lt 0$. Then 
$\lambda_1(m)$ is simple and any associated eigenfunction is one signed in Ω.
We call 
$\lambda_1(m)$ the principal eigenvalue of problem (1). Throughout the paper we will denote by um the unique positive eigenfunction of both Gm (relative to 
$\mu_1(m)$) and problem (1) (relative to 
$\lambda_1(m)$), normalized by
\begin{equation}
\|u_m\|_{V_m(\Omega)}=1,
\end{equation}which is equivalent to
\begin{equation}
\int_\Omega m u_m^2 \, dx= \mu_1(m)=\cfrac{1}{\lambda_1(m)}\,.
\end{equation} By standard regularity theory (see [Reference Gilbarg and Trudinger27]), 
$u_m\in W^{2,2}_{\rm loc}(\Omega)\cap C^{1,
\beta}(\Omega)$ for all 
$0 \lt \beta \lt 1$.
As last comment, we observe that 
$\mu_1(m)$ is homogeneous of degree 1, i.e.
\begin{equation}\mu_1(\alpha m)= \alpha \mu_1(m) \quad \forall\, \alpha \gt 0.\end{equation}This follows immediately from (26).
3. Optimization of
$\lambda_1(m)$
This section is devoted to the study of the optimization of 
$\lambda_1(m)$. For this purpose, we need some qualitative properties of 
$\mu_1(m)=1/\lambda_1(m)$ with respect to m. We begin by proving the continuity of 
$\mu_1(m)$ (actually, of all of the eigenvalues of the operator Gm defined in (19)) and then showing its convexity and Gâteaux differentiability. The structure of the proofs follows the ideas contained in [Reference Anedda, Cuccu and Frassu3] in the case of the fractional Laplacian and Dirichlet boundary conditions. Here, we deal with Neumann boundary conditions which require, especially in proving continuity, a more sophisticated argument involving the projection Pm.
Finally, we examine the minimization and maximization of 
$\lambda_1(m)$.
We introduce the following convex subset of 
$L^{\infty}(\Omega)$
\begin{equation*}
L^\infty_ \lt (\Omega)=\left\{m\in L^\infty (\Omega): \int_\Omega m\,dx \lt 0\right\}. \end{equation*} Observe that, by Proposition 5, 
$\mu_k(m)$ and um (the unique positive eigenfunction of 
$\mu_1(m)$ of problem (1) normalized as in (28)) are well defined only when 
$|\{m \gt 0\}| \gt 0$. We extend them to the whole set 
$L^\infty_ \lt (\Omega)$ by putting
\begin{equation}
\widetilde{\mu}_k(m)=\begin{cases} \mu_k(m) \quad & \text{if } |\{m \gt 0\}| \gt 0\\
0 & \text{if } |\{m \gt 0\}|=0\end{cases}
\end{equation}and
\begin{equation}
\widetilde{u}_m=\begin{cases}u_m \quad & \text{if } |\{m \gt 0\}| \gt 0\\
0 & \text{if } |\{m \gt 0\}|=0.\end{cases}
\end{equation}Remark 1. Note that 
$\widetilde{\mu}_k(m)=0$ if and only if 
$|\{m \gt 0\}|=0$ and, in this circumstance, the inequality
\begin{equation}\sup_{F_k\subset V_m(\Omega)}
\min_{f\in F_k\atop f\neq 0}
\cfrac{\langle G_m(f), f \rangle_{V_m(\Omega)} }{\|f\|^2_{V_m(\Omega)}} \,\leq 0
\end{equation} holds, where Fk varies among all the k-dimensional subspaces of 
$V_m(\Omega)$. Moreover, from (30), we have 
$\widetilde{\mu}_1(\alpha m)=\alpha \widetilde{\mu}
_1(m)$ for every 
$\alpha\geq 0$.
Lemma 2. Let 
$m\in L_ \lt ^\infty(\Omega)$ and Em be the linear operator (14). Then, the map 
$m\mapsto E_m$ is sequentially weakly* continuous from 
$L_ \lt ^\infty(\Omega)$ to 
$\mathcal{L}(H^1(\Omega),H^1(\Omega)) $ endowed with the norm topology.
Proof. Let 
$\{m_i\}$ be a sequence which weakly* converges to m in 
$L^\infty_ \lt (\Omega)$. Being 
$\{m_i\}$ bounded in 
$L^\infty(\Omega)$, there exists a constant M > 0 such that
\begin{equation}
\|m\|_{L^\infty(\Omega)}\leq M\quad \text{and } \quad \|m_i\|_{L^\infty(\Omega)}\leq M
\quad \forall\, i.
\end{equation} We begin by proving that 
$E_{m_i}(f)$ tends to 
$E_m(f)$ in 
$H^1(\Omega)$ for any fixed 
$f\in H^1(\Omega)$. Recalling (i) of Proposition 4, we put 
$u_i=E_{m_i}(f)= G_{m_i}(P_{m_i}(f))$ and 
$u=E_{m}(f)= G_{m}(P_{m}(f))$.
First, we show that 
$P_m(u_i)$ weakly converges to u in 
$V_m(\Omega)$; indeed, by (15) we have
\begin{equation*}
\langle u_i,\varphi\rangle_{V_{m_i}(\Omega)}=
\langle m_if,\varphi\rangle_{L^2(\Omega)}\quad \forall\varphi \in V_{m_i}(\Omega).
\end{equation*}By Lemma 1 and (iii) of Proposition 1, we find
\begin{equation}
\langle\nabla u_i,\nabla\varphi\rangle_{L^2(\Omega)}= \langle m_i P_{m_i}(f),
\varphi\rangle_{L^2(\Omega)} \quad\forall\, \varphi\in
H^1(\Omega).
\end{equation}Similarly, for u we have
\begin{equation}
\langle\nabla u,\nabla\varphi\rangle_{L^2(\Omega)}= \langle m P_{m}(f),\varphi\rangle_{L^2(\Omega)} \quad\forall\, \varphi\in H^1(\Omega).
\end{equation} Taking 
$\varphi\in V_m(\Omega)$ in (35) and (36) and by using (iii) of Proposition 1, we find
\begin{equation}
\langle P_m(u_i),\varphi\rangle_{V_m(\Omega)}=
\langle m_iP_{m_i}(f),\varphi\rangle_{L^2(\Omega)}\quad \forall\varphi \in V_m(\Omega)
\end{equation}and
\begin{equation}
\langle u,\varphi\rangle_{V_m(\Omega)}=
\langle mP_{m}(f),\varphi\rangle_{L^2(\Omega)}\quad \forall\varphi \in V_m(\Omega).
\end{equation}Subtracting (38) from (37), we get
\begin{equation}
\langle P_m(u_i)-u,\varphi\rangle_{V_m(\Omega)}=
\langle m_iP_{m_i}(f)-m P_m(f),\varphi\rangle_{L^2(\Omega)}\quad \forall\varphi \in V_m(\Omega).
\end{equation} As a consequence of the weak* convergence of mi to m in 
$L^\infty(\Omega)$, letting 
$i\to
\infty$ we obtain 
$\int_\Omega m_i f\varphi \,dx \to \int_\Omega m f\varphi \,dx$, 
$\int_\Omega m_i
f\,dx \to \int_\Omega m f\,dx$, 
$\int_\Omega m_i\varphi \,dx \to \int_\Omega m\varphi \,dx$ and 
$
\int_\Omega m_i\,dx \to \int_\Omega m\,dx$, which imply that the right hand term goes to zero, thus 
$P_m(u_i)$ weakly converges to u in 
$V_m(\Omega)$. By exploiting the continuity of the inclusion 
$V_m(\Omega)\hookrightarrow H^1(\Omega)$ and the compactness of 
$H^1(\Omega)\hookrightarrow L^2(\Omega)$, we deduce that 
$P_m(u_i)$ weakly converges to u in 
$H^1(\Omega)$ and strongly in 
$L^2(\Omega)$. Putting 
$\varphi=P_m(u_i)-u$ in (39) we get
\begin{equation}
\begin{aligned}
\| P_m(u_i)-u\|^2_{V_m(\Omega)}&=
\langle m_iP_{m_i}(f)-m P_m(f),P_m(u_i)-u\rangle_{L^2(\Omega)}\\
&\leq\left(\| m_iP_{m_i}(f)\|_{L^2(\Omega))}+\| mP_{m}(f)\|_{L^2(\Omega)}\right)\| P_m(u_i)-u\|_{L^2(\Omega)}.
\end{aligned}
\end{equation}By using (34), (6) and (8), we find
\begin{equation}
\| mP_{m}(f)\|_{L^2(\Omega)}\leq\,M\| P_{m}(f)\|_{L^2(\Omega)}
\leq \, M C_1(m)\|f\|_{L^2(\Omega)}.
\end{equation}Similarly, we have
\begin{equation}
\| m_iP_{m_i}(f)\|_{L^2(\Omega)}\leq M C_1(m_i)\|f\|_{L^2(\Omega)}.
\end{equation}By the weak* convergence of mi to m we can assume
\begin{equation*}
\left|\int_\Omega m_i\,dx\right| \gt \frac{\left|\int_\Omega m\,dx\right|}{2}
\end{equation*}for i large enough. Therefore
\begin{equation}
C_1(m_i)\leq 1+ \frac{M}{\left|\int_\Omega m_i\, dx \right|}\, |\Omega|\leq 1+ \frac{2M}{\left|\int_\Omega m\, dx \right|}\, |\Omega|
\end{equation}and, trivially
\begin{equation}
C_1(m)\leq 1+ \frac{2M}{\left|\int_\Omega m\, dx \right|}\, |\Omega|.
\end{equation}For the sake of simplicity we put
\begin{equation}
D(m,M)=1+ \frac{2M}{\left|\int_\Omega m\, dx \right|}\, |\Omega|.
\end{equation}Then, by (40), (41), (42), (43), (44) and (45) we find
\begin{equation*}
\| P_m(u_i)-u\|_{V_m(\Omega)}^2\leq 2MD(m,M)\|f\|_{L^2(\Omega)} \| P_m(u_i)-u\|_{L^2(\Omega)},
\end{equation*} from which the convergence of 
$P_{m}(u_i)$ to u in 
$V_m(\Omega)$ follows. The next step shows that, actually, ui strongly converges to u in 
$L^2(\Omega)$. Indeed, by using Definition 1, (vi) of Proposition 1, (6), (43) and (45), we have
\begin{equation*}
\begin{aligned}
\|u_i-u\|_{L^2(\Omega)}=&\left \| -\frac{\int_\Omega m_iu\,dx}{\int_\Omega m_i\, dx}+P_{m_i}\left(P_m(u_i)-u\right)\right\|_{L^2(\Omega)}\\
\leq & \left \| \frac{\int_\Omega m_iu\,dx}{\int_\Omega m_i\, dx}\right\|_{L^2(\Omega)}+ \left \|P_{m_i}\left(P_m(u_i)-u\right)\right\|_{L^2(\Omega)}\\
\leq &\left | \frac{\int_\Omega m_iu\,dx}{\int_\Omega m_i\, dx}\right| |\Omega|^{1/2}+
D(m,M)\left\|P_m(u_i)-u\right\|_{L^2(\Omega)},
\end{aligned}
\end{equation*} which goes to zero because 
$\int_\Omega m_i
u\,dx/\int_\Omega m_i \,dx \to \int_\Omega m
u\,dx/\int_\Omega m \,dx=0$ (being 
$u\in V_m(\Omega)$) and 
$P_m(u_i)\to u$ in 
$L^2(\Omega)$. Moreover, by (iv) of Proposition 1, we have
\begin{equation*}
\| u_i-u\|_{H^1(\Omega)}=\left(\|u_i-u\|^2_{L^2(\Omega)}+\| P_m(u_i)-u\|_{V_m(\Omega)}^2\right)^{1/2}
\end{equation*} and then ui converges to u in 
$H^1(\Omega)$. Summarizing, for every 
$f\in H^1(\Omega)$ we have
\begin{equation*}
\|E_{m_i}(f) - E_m (f)\|_{H^1(\Omega)}\to 0\quad\text{for }i\to\infty.
\end{equation*} Now, for fixed i, let 
$\{f_{i,j}\}$, 
$j=1,2, 3,\ldots$, be a maximizing sequence of
\begin{equation*}\sup_{g\in H^1(\Omega)\atop \|g\|_{H^1(\Omega)}\leq 1}
\|E_{m_i}(g) - E_{m}(g)\|_{H^1(\Omega)}=\|E_{m_i}-E_m\|_{\mathcal{L}(H^1(\Omega), H^1(\Omega))} .\end{equation*} Then, being 
$\|f_{i,j}\|_{H^1(\Omega)}\leq 1$, we can extract a subsequence (still denoted by 
$\{f_{i,j}\}$) weakly convergent to some 
$f_i\in H^1(\Omega)$. Since the operators 
$E_{m_i}$ and Em restricted to 
$H^1(\Omega)$ are compact (see iii) of Proposition 4), it follows that 
$E_{m_i}(f_{i,j})$ converges to 
$E_{m_i}(f_{i})$ and 
$E_{m}(f_{i,j})$ converges to 
$E_{m}(f_{i})$ strongly in 
$V_m(\Omega)$ and then in 
$H^1(\Omega)$ as j goes to 
$\infty$. Thus we find
\begin{equation*} \|E_{m_i}-E_m\|_{\mathcal{L}(H^1(\Omega), H^1(\Omega))}
=\|E_{m_i}(f_i) - E_m(f_i)\|_{H^1(\Omega)}.\end{equation*} This procedure yields a sequence 
$\{f_i\}$ in 
$H^1(\Omega)$ such that 
$\|f_{i}\|_{H^1(\Omega)}\leq 1$ for all i. Then, up to a subsequence, we can assume that 
$\{f_i\}$ weakly converges to a function 
$f\in H^1(\Omega)$ and (by the compactness of the inclusion 
$H^1(\Omega)\hookrightarrow L^2(\Omega)$) strongly in 
$L^2(\Omega)$. By using (12), (43), (44), (45), (17) and (34) we find
\begin{align*} & \|E_{m_i}- E_m\|_{\mathcal{L}(H^1(\Omega), H^1(\Omega))}=
\|E_{m_i}(f_i) - E_m(f_i)\|_{H^1(\Omega)}\\
& \leq \|E_{m_i}(f) - E_m(f)\|_{H^1(\Omega)} + \|E_{m_i}(f_i-f)\|_{H^1(\Omega)} +\|E_m(f_i-f)\|_
{H^1(\Omega)}\\
& \leq \|E_{m_i}(f) - E_m(f)\|_{H^1(\Omega)}
+ \left(C^2\cdot C_1^2(m_i)+1\right)^{1/2}\|
E_{m_i}(f_i-f)\|_{V_{m_i}(\Omega)}\\
&+\left(C^2\cdot C_1^2(m)+1\right)^{1/2}\|E_m(f_i-f)\|_
{V_{m}(\Omega)}\\
&\leq \|E_{m_i}(f) - E_m(f)\|_{H^1(\Omega)}\\
& +\left(C^2\cdot D^2(m,M)+1\right)^{1/2}\left(\|E_{m_i}\|
_{\mathcal{L}(L^2(\Omega), V_{m_i}(\Omega))}+\|E_{m}\|_{\mathcal{L}(L^2(\Omega),
V_m(\Omega))} \right) \\&
\|f_i-f\|_{L^2(\Omega)} \leq \|E_{m_i}(f) - E_m(f)\|_{H^1(\Omega)}\\
& +\left(C^2\cdot D^2(m,M)+1\right)^{1/2} \left (C\cdot C_1(m_i)\|m_i\|_{L^\infty(\Omega)}
+C\cdot C_1(m)\|m\|_{L^\infty(\Omega)}
\right)\\
& \|f_i-f\|_{L^2(\Omega)}\leq \|E_{m_i}(f) - E_m(f)\|_{H^1(\Omega)}+2CMD(m,M) \\ &\left (C^2\cdot D^2(m,M)
+1\right)^{1/2}
\|f_i-f\|_{L^2(\Omega)}.
\end{align*} Therefore 
$E_{m_i}$ converges to Em in the operator norm.
Remark 2. We note that the previous lemma still holds replacing 
$L_ \lt ^\infty(\Omega)$ by the set of 
$L^\infty(\Omega)$ such that 
$\int_\Omega m\;dx\neq 0$.
Lemma 3. Let 
$m\in L_ \lt ^\infty(\Omega)$, 
$\widetilde{\mu}_k(m)$ as defined in (31) for 
$k=1,2,3,\ldots$ and 
$\widetilde{u}_m$ as in (32). Then
i) the map 
$m\mapsto \widetilde{\mu}_k(m)$ is sequentially weakly* continuous in 
$L_ \lt ^\infty(\Omega)$;
(ii) the map 
$m\mapsto \widetilde{\mu}_1(m)\widetilde{u}_m$ is sequentially weakly* continuous from 
$L_ \lt ^{\infty}(\Omega)$ to 
$H^1(\Omega)$ (endowed with the norm topology). In particular, for any sequence 
$\{m_i\}$ weakly* convergent to 
$m\in L_ \lt ^\infty(\Omega)$ with 
$\widetilde{\mu}_1(m) \gt 0$, then 
$\{\widetilde{u}_{m_i}\}$ converges to 
$\widetilde{u}_{m}$ in 
$H^1(\Omega)$.
Proof. i) Let 
$\{m_i\}$ be a sequence which weakly* converges to m in 
$L^\infty_ \lt (\Omega)$. Being 
$\{m_i\}$ bounded in 
$L^\infty(\Omega)$, there exists a constant M > 0 such that (34) holds. We will show that
\begin{equation}
|\widetilde{\mu}_k(m_i)- \widetilde{\mu}_k(m)|\leq D(m,M)(C^2\cdot D^2(m,M)+1)^{1/2} \|E_{m_i}-E_m\|_{\mathcal{L}(H^1(\Omega), H^1(\Omega))},
\end{equation} where 
$D(m,M)$ is the constant in (45). By Lemma 2 the claim follows. We split the argument in three cases.
Case 1. Let i be fixed and assume 
$\widetilde{\mu}_k(m_i)$, 
$\ \widetilde{\mu}_k(m) \gt 0$.
Following the idea in [Reference Henrot29, Theorem 2.3.1] and by means of the Fischer’s Principle (20) we have
\begin{equation*} \begin{aligned}
\widetilde{\mu}_k(m_i)- \widetilde{\mu}_k(m) &= \max_{F_k\subset V_{m_i}(\Omega)}
\min_{f\in F_k\atop f\neq 0}
\cfrac{\langle G_{m_i}(f), f \rangle_{V_{m_i}(\Omega)} }{\|f\|^2_{V_{m_i}(\Omega)}} \nonumber\\
&\quad - \max_{F_k\subset V_m(\Omega)}
\min_{f\in F_k\atop f\neq 0}
\cfrac{\langle G_m (f), f \rangle_{V_m(\Omega)} }{\|f\|^2_{V_m(\Omega)}} \,\\
& \leq
\min_{f\in \overline{F_k}\atop f\neq 0}
\cfrac{\langle G_{m_i} (f), f \rangle_{V_{m_i}(\Omega)} }{\|f\|^2_{V_{m_i}(\Omega)}} \,
-
\min_{f\in P_m (\overline{F_k})\atop f\neq 0}
\cfrac{\langle G_m (f), f \rangle_{V_m(\Omega)} }{\|f\|^2_{V_m(\Omega)}} \,\\
& \leq
\cfrac{\langle G_{m_i}(\overline{f}), \overline{f} \rangle_{V_{m_i}(\Omega)} }{\|\overline{f}\|^2_{V_{m_i}(\Omega)}} \,-
\cfrac{\langle G_m (P_m(\overline{f})), P_m(\overline{f}) \rangle_{V_m(\Omega)} }{\|P_m(\overline{f})\|^2_{V_m(\Omega)}}\, ,
\end{aligned}
\end{equation*} where 
$\overline{F_k}$ is a k-dimensional subspace of 
$V_{m_i}(\Omega)$ such that
\begin{equation*}\max_{F_k\subset V_{m_i}(\Omega)}
\min_{f\in F_k\atop f\neq 0}
\cfrac{\langle G_{m_i}(f), f \rangle_{V_{m_i}(\Omega)} }{\|f\|^2_{V_{m_i}(\Omega)}} \, =\min_{f\in \overline{F_k}\atop f\neq 0}
\cfrac{\langle G_{m_i} (f), f \rangle_{V_{m_i}(\Omega)} }{\|f\|^2_{V_{m_i}(\Omega)}} \,\end{equation*} (note that, by vii) of Proposition 1, 
$P_m (\overline{F_k})$ is a k-dimensional subspace of 
$V_{m}(\Omega)$) and 
$\overline{f}$ is a function in 
$\overline{F_k}$ such that
\begin{equation*} \min_{f\in P_m(\overline{F_k})\atop f\neq 0}
\cfrac{\langle G_m (f), f \rangle_{V_m(\Omega)} }{\|f\|^2_{V_m(\Omega)}} \,=
\cfrac{\langle G_m(P_m(\overline{f})), P_m(\overline{f}) \rangle_{V_m(\Omega)} }
{\|P_m(\overline{f})\|^2_{V_m(\Omega)}} \,.
\end{equation*} By (iv) of Proposition 1 
$\nabla P_m(\overline{f})=\nabla\overline{f}$ and 
$\nabla G_m(P_m(\overline{f}))=\nabla P_{m_i}(G_m(P_m(\overline{f})))$ hold, thus we have
\begin{equation*}
\begin{aligned}
\widetilde{\mu}_k(m_i)- \widetilde{\mu}_k(m) & \leq
\cfrac{\langle G_{m_i}(\overline{f}), \overline{f} \rangle_{V_{m_i}(\Omega)} }{\|\overline{f}\|^2_{V_{m_i}(\Omega)}} \,-
\cfrac{\langle P_{m_i}(G_m (P_m(\overline{f}))),\overline{f}\rangle_{V_{m_i}(\Omega)} }{\|\overline{f}\|^2_{V_{m_i}(\Omega)}} \,\\
& = \cfrac{\langle (G_{m_i}-P_{m_i}\circ G_m\circ P_m)(\overline{f}), \overline{f} \rangle_{V_{m_i}(\Omega)} }{\|\overline{f}\|^2_{V_{m_i}(\Omega)}} \,\\
&\leq
\cfrac{\|(G_{m_i}-P_ {m_i}\circ G_m\circ P_m)(\overline{f})\|_{V_{m_i}(\Omega)}}{\|\overline{f}\|_{V_{m_i}(\Omega)}}\,.
\end{aligned}
\end{equation*}Then, taking into account the identities (recall iii) of Proposition 1)
\begin{equation*}G_{m_i}-P_{m_i}\circ G_m\circ P_m=P_{m_i}\circ G_{m_i}\circ P_{m_i}-P_{m_i}\circ G_m\circ P_m=P_{m_i}\circ(G_{m_i}\circ P_{m_i}-G_m\circ P_m),\end{equation*}(12), (7), (8), (43), (45) and (i) of Proposition 4 we find
\begin{equation*}
\begin{aligned}
\widetilde{\mu}_k(m_i)- \widetilde{\mu}_k(m) &\leq\cfrac{\| P_{m_i}\circ(G_{m_i}\circ P_{m_i}-G_m\circ P_m)(\overline{f})\|_{V_{m_i}(\Omega)}}{\|\overline{f}\|_{V_{m_i}(\Omega)}}\\
&\leq\cfrac{\| P_{m_i}\circ(G_{m_i}\circ P_{m_i}-G_m\circ P_m)(\overline{f})\|_{H^1(\Omega)}}{\|\overline{f}\|_{V_{m_i}(\Omega)}}\\
&\leq (C^2\cdot C_1^2(m_i)+1)^{1/2}\|P_{m_i}\|_{\mathcal{L}(H^1(\Omega), H^1(\Omega))}\\
&\quad \times \cfrac{\|(G_{m_i}\circ P_{m_i}-G_m\circ P_m)(\overline{f})\|_{H^1(\Omega)} }{\|\overline{f}\|_{H^1(\Omega)}}\\
&\leq C_1(m_i)(C^2\cdot C_1^2(m_i)+1)^{1/2}\|G_{m_i}\circ P_{m_i}\\
&\quad -G_m\circ P_m\|
_{\mathcal{L}(H^1(\Omega), H^1(\Omega))}\\
&\leq D(m,M)(C^2\cdot D^2(m,M)+1)^{1/2} \|E_{m_i}-E_m\|_{\mathcal{L}(H^1(\Omega), H^1(\Omega))}.
\end{aligned}
\end{equation*}Interchanging the role of mi and m and replacing (43) by (44), we also have
\begin{equation*}
\widetilde{\mu}_k(m)- \widetilde{\mu}_k(m_i)\leq D(m,M)(C^2\cdot D^2(m,M)+1)^{1/2} \|E_{m_i}-E_m\|_{\mathcal{L}(H^1(\Omega), H^1(\Omega))}
\end{equation*}and finally (46).
Case 2. Let i be fixed and assume 
$\widetilde{\mu}_k(m_i) \gt 0$, 
$ \widetilde{\mu}_k(m)=0$ (and similarly in the case 
$\widetilde{\mu}_k(m) \gt 0$, 
$ \widetilde{\mu}_k(m_i)=0$).
Note that in this case (33) holds for the weight function m. Then the previous argument still applies provided that we replace the first step of the inequality chain by
\begin{align*} |\widetilde{\mu}_k(m_i)- \widetilde{\mu}_k(m) |&=
\widetilde{\mu}_k(m_i) \leq \max_{F_k\subset V_{m_i}(\Omega)}
\min_{f\in F_k\atop f\neq 0}
\cfrac{\langle G_{m_i}(f), f \rangle_{V_{m_i}(\Omega)} }{\|f\|^2_{V_{m_i}(\Omega)}} \\
&\quad - \sup_{F_k\subset V_m(\Omega)}
\min_{f\in F_k\atop f\neq 0}
\cfrac{\langle G_m (f), f \rangle_{V_m(\Omega)} }{\|f\|^2_{V_m(\Omega)}} \,.
\end{align*} Case 3. 
$\widetilde{\mu}_k(m_i)= \widetilde{\mu}_k(m)=0$.
In this case, (46) is obvious.
Therefore statement (i) is proved.
(ii) Let 
$\{m_i\}$ be such that mi is weakly
$^*$ convergent to 
$m\in L_ \lt ^\infty(\Omega)$. By using (12), (43) and (45), for i sufficiently large, we have
\begin{equation*}\|\widetilde{u}_{m_i}\|_{H^1(\Omega)}
\leq\left(C^2\cdot D^2(m,M)+1\right)^{1/2},\end{equation*} up to a subsequence we can assume that 
$\widetilde{u}_{m_i}$ is weakly convergent to 
$z
\in H^1(\Omega)$, strongly in 
$L^2(\Omega)$ and pointwisely a.e. in Ω.
First suppose 
$\widetilde{\mu}_1(m)=0$. Then, by (i), 
$\widetilde{\mu}_1(m_i) \widetilde{u}_{m_i}$ weakly converges in 
$H^1(\Omega)$ to 
$\widetilde{\mu}_1(m)z=0= \widetilde{\mu}_1(m) \widetilde{u}_{m}$. Moreover, 
$\|\widetilde{\mu}_1(m_i) \widetilde{u}_{m_i}\|_{H^1(\Omega)}
=\widetilde{\mu}_1(m_i)\| \widetilde{u}_{m_i}\|_{H^1(\Omega)}$ tends to 
$0=\|\widetilde{\mu}_1(m) \widetilde{u}_{m}\|_{H^1(\Omega)}$. Therefore 
$\widetilde{\mu}_1(m_i) \widetilde{u}_{m_i}$ strongly converges to 
$\widetilde{\mu}_1(m) \widetilde{u}_{m}$ in 
$H^1(\Omega)$.
Next, consider the case 
$\widetilde{\mu}_1(m) \gt 0$. By (i) we have 
$\widetilde{\mu}_1(m_i) \gt 0$ for all i large enough. This implies 
$\widetilde{\mu}_1(m_i)
=\frac{1}{\lambda_1(m_i)}\, $ and 
$\widetilde{u}_{m_i}= u_{m_i}$. Positiveness and pointwise convergence of 
$u_{m_i}$ to z imply 
$z\geq 0$ a.e. in Ω. Moreover, by (29) we have
\begin{equation*} \int_\Omega m_i u^2_{m_i} \, dx =\cfrac{1}{\lambda_1(m_i)}\, \end{equation*}and by (i), passing to the limit, we find
\begin{equation*} \int_\Omega m z^2 \, dx =\cfrac{1}{\lambda_1(m)}\, ,\end{equation*} which implies z ≠ 0. By using (25) for 
$u_{m_i}$ we have
\begin{equation*}\langle \nabla u_{m_i}, \nabla\varphi\rangle_{L^2(\Omega)} =
\lambda_1(m_i) \langle m_i u_{m_i}, \varphi\rangle_{L^2(\Omega)} = \lambda_1(m_i)
\int_\Omega m_i u_{m_i} \varphi \, dx \quad \forall\, \varphi \in H^1(\Omega),
\end{equation*} and, letting i to 
$\infty$, we deduce 
$z=u_m$. By (i) 
$\mu_1(m_i) u_{m_i}$ weakly converges in 
$H^1(\Omega)$ to 
$\mu_1(m)u_{m}$ and 
$\|\mu_1(m_i)u_{m_i}\|_{H^1(\Omega)}
=\mu_1(m_i)$ tends to 
$\mu_1(m) =\|\mu_1(m) u_{m}\|_{H^1(\Omega)}$. Hence 
$\mu_1(m_i) u_{m_i}$ strongly converges to 
$\mu_1(m)u_{m}$ in 
$H^1(\Omega)$. The last claim is immediate provided one observes that 
$\widetilde{\mu}_1(m) \gt 0$ implies 
$\widetilde{\mu}_1(m_i) \gt 0$ for all i large enough.
Lemma 4. Let 
$m, q\in L_ \lt ^\infty(\Omega)$, 
$\widetilde{\mu}_1(m)$ be defined as in (31) for k = 1. Then
(i) the map 
$m\mapsto \widetilde{\mu}_1(m)$ is convex on 
$L_ \lt ^\infty(\Omega) $;
(ii) if m and q are linearly independent and 
$ \widetilde{\mu}_1(m), \widetilde{\mu}_1(q) \gt 0$, then
\begin{equation*} \widetilde{\mu}_1(tm+(1-t)q) \lt t \widetilde{\mu}_1(m)+(1-t) \widetilde{\mu}_1(q)
\end{equation*} for all 
$0 \lt t \lt 1$.
Proof. (i) The Fischer’s Principle (20) and (33) both for k = 1 yield
\begin{equation}
\sup_{f\in V_m(\Omega) \atop f\neq 0}
\cfrac{\int_\Omega mf^2\,dx}{\int_\Omega|\nabla f|^2\,dx}\,\leq \widetilde{\mu}_1(m)
\end{equation} for every 
$m \in L_ \lt ^\infty(\Omega)$. Moreover, if 
$\widetilde{\mu}_1(m) \gt 0$, then the equality sign holds and the supremum is attained when f is an eigenfunction of 
$\widetilde{\mu}_1(m)=\mu_1(m)$. Let 
$m, q\in L_ \lt ^\infty(\Omega)$, 
$0\leq t\leq 1$. We show that
\begin{equation}
\widetilde{\mu}_1(tm + (1-t)q)\leq t \widetilde{\mu}_1 (m) +(1-t) \widetilde{\mu}_1 (q).
\end{equation} If 
$\widetilde{\mu}_1(tm + (1-t)q)=0$, (48) is obvious. Suppose 
$\widetilde{\mu}_1(tm + (1-t)q) \gt 0$. Then, for all 
$f\in V_{tm + (1-t)q}(\Omega)$, f ≠ 0, we have
\begin{equation}
\begin{aligned}
\cfrac{\int_\Omega (tm+(1-t)q)f^2\,dx}{\int_\Omega|\nabla f|^2\,dx}
\,=&\,t\,\cfrac{\int_\Omega mf^2\,dx}{\int_\Omega|\nabla f|^2\,dx}\,
+(1-t)\, \cfrac{\int_\Omega qf^2\,dx}{\int_\Omega|\nabla f|^2\,dx}\\
\leq &\,t\,\cfrac{\int_\Omega mf^2\,dx-\frac{\left(\int_\Omega mf\,dx\right)^2}{\int_\Omega m\,dx}}{\int_\Omega|\nabla f|^2\,dx}\\
&\quad +(1-t)\,\cfrac{\int_\Omega qf^2\,dx-\frac{\left(\int_\Omega qf\,dx\right)^2}{\int_\Omega q\,dx}}{\int_\Omega|\nabla f|^2\,dx}\\
=&\,t\,\cfrac{\int_\Omega m(P_m(f))^2\,dx}{\int_\Omega|\nabla P_m(f)|^2\,dx}\,
+(1-t)\, \cfrac{\int_\Omega q(P_q(f))^2\,dx}{\int_\Omega|\nabla P_q(f)|^2\,dx}\\
\leq&\, t\widetilde{\mu}_1 (m) +(1-t) \widetilde{\mu}_1 (q),\\
\end{aligned}
\end{equation}where we used (iv) of Proposition 1 and (47) for m and q. Taking the supremum in the left-hand term of (49) and using (47) again with equality sign, we find (48).
(ii) Arguing by contradiction, we suppose that equality holds in (48). We will conclude that m and q are linearly dependent. Equality sign in (48) implies 
$\widetilde{\mu}_1(tm + (1-t)q) \gt 0$, then (by (47)) equalities also occur in (49) with 
$f=u=u_{tm+(1-t)q}$. We get 
$\int_\Omega mu\,dx=\int_\Omega qu\,dx=0$, thus 
$u\in V_m(\Omega)\cap V_q(\Omega)$, and then, by (iii) of Proposition 1,
\begin{equation*}\cfrac{\int_\Omega mu^2\,dx}{\int_\Omega|\nabla u|^2\,dx}\, =\widetilde{\mu}_1(m) \,\quad
\text{and } \quad\cfrac{\int_\Omega qu^2\,dx}{\int_\Omega|\nabla u|^2\,dx}\,
=\widetilde{\mu}_1(q).\end{equation*} The simplicity of the principal eigenvalue, the positiveness of u and the normalization (28) imply that 
$u=u_m=u_q$. By using (25) with 
$\lambda=\frac{1}{\tilde{\mu}_1(m)}$ and 
$\lambda=\frac{1}{\tilde{\mu}_1(q)}$ we have
\begin{equation*} \langle \nabla u, \nabla\varphi \rangle_{L^2(\Omega)} =\cfrac{1}{\widetilde{\mu}_1(m)}\,
\langle m u, \varphi\rangle_{L^2(\Omega)} \quad \forall\, \varphi \in H^1(\Omega) ,
\end{equation*}and
\begin{equation*}\langle\nabla u, \nabla\varphi\rangle_{L^2(\Omega)} = \cfrac{1}{\widetilde{\mu}_1(q)}\,
\langle q u, \varphi\rangle_{L^2(\Omega)} \quad \forall\, \varphi\in H^1(\Omega)\end{equation*}respectively. Taking the difference of these identities we find
\begin{equation*}\left\langle \left( \cfrac{m}{\widetilde{\mu}_1(m)}\, -\cfrac{q}{\widetilde{\mu}_1(q)}\, \right) u, \varphi\right\rangle_{L^2(\Omega)} =0 \quad \forall\, \varphi\in H^1(\Omega), \end{equation*} which gives 
$m\widetilde{\mu}_1(q)-q\widetilde{\mu}_1(m)=0$ a.e. in Ω, i.e. m and q are linearly dependent.
Corollary 2. Let 
$m_0\in L_ \lt ^\infty(\Omega)$, 
$\widetilde{\mu}_1(m)$ be defined as in (31) for k = 1 and 
$\overline{\mathcal{G}(m_0)}$ the weak* closure in 
$L^\infty(\Omega)$ of the class of rearrangements 
$\mathcal{G}(m_0)$ introduced in Definition 7. Then the map 
$m\mapsto \widetilde{\mu}_1(m)$ is convex but not strictly convex on 
$\overline{\mathcal{G}(m_0)}$.
Proof. By (ii) of Proposition 15 and (i) of Corollary 3, we have that 
$
\overline{\mathcal{G}(m_0)}$ is convex and 
$\overline{\mathcal{G}(m_0)}\subset
L_ \lt ^\infty(\Omega)$. Then, by Lemma 4, the map 
$m\mapsto \widetilde{\mu}_1(m)$ is convex on 
$\overline{\mathcal{G}(m_0)}$.
Applying Proposition 13, we find that the constant function 
$c=\frac{1}{|\Omega|}
\int_\Omega m_0\,dx$ is in 
$\overline{\mathcal{G}(m_0)}$. By convexity of 
$
\overline{\mathcal{G}(m_0)}$, 
$tm_0+(1-t)c\in \overline{\mathcal{G}(m_0)}$ for every 
$t\in [0,1]$. From the inequality
\begin{equation*}tm_0+(1-t)c\leq t\|m_0\|_{L^\infty(\Omega)}+(1-t)c\quad\text{a.e. in }\Omega,\end{equation*}we obtain
\begin{equation*}tm_0+(1-t)c\leq 0\quad\text{a.e. in }\Omega \quad \forall\, t\leq \frac{c}{c-\|m_0\|
_{L^\infty(\Omega)}}. \end{equation*} Note that 
$c/\!\!\left(c-\|m_0\|_{L^\infty(\Omega)}\right)\in (0,1)$. Therefore, by (31), we conclude that 
$\widetilde{\mu}_1(m)=0$ for any m in the line segment, contained in 
$\overline{\mathcal{G}(m_0)}$, that joins c and
\begin{equation*}
\frac{c}{c-\|m_0\|_{L^\infty(\Omega)}} m_0+\left(1-\frac{c}{c-\|m_0\|_{L^\infty(\Omega)}}
\right)c=\frac{\|m_0\|_{L^\infty(\Omega)}-m_0}{\|m_0\|_{L^\infty(\Omega)}-c}c.\end{equation*} This shows that the map 
$m\mapsto \widetilde{\mu}_1(m)$ is not strictly convex.
For the definitions and some basic results on the Gâteaux differentiability we refer the reader to [Reference Ekeland and Témam24].
Lemma 5. Let 
$m\in L_ \lt ^\infty(\Omega)$, 
$\widetilde{\mu}_1(m)$ be defined as in (31) for k = 1 and um denote the relative unique positive eigenfunction of problem (1) normalized as in (28). Then, the map 
$m\mapsto \widetilde{\mu}_1(m)$ is Gâteaux differentiable at any m such that 
$ \widetilde{\mu}_1(m) \gt 0$, with Gâteaux differential equal to 
$u_m^2$. In other words, for every direction 
$v\in L^\infty(\Omega)$ we have
\begin{equation} \widetilde{\mu}_1'(m; v) =\int_\Omega u_m^2 v\, dx. \end{equation}Proof. Let us compute
\begin{equation*}\lim_{t\to 0} \cfrac{\widetilde{\mu}_1(m+ t v)-
\widetilde{\mu}_1(m)}{t}\, .\end{equation*} Note that 
$m+tv\in L_ \lt ^\infty(\Omega)$ for 
$|t|$ sufficiently small and by (i) of Lemma 3, 
$\widetilde{\mu}_1(m+ t v)$ converges to 
$\widetilde{\mu}_1(m)$ as t goes to zero for any 
$m\in L_ \lt ^\infty(\Omega)$ and 
$v\in L^\infty(\Omega)$. Therefore, 
$\widetilde{\mu}_1(m+ t v) \gt 0$ for 
$|t|$ small enough. The eigenfunctions um and 
$u_{m+tv}$ satisfy (see (25))
\begin{equation*}\widetilde{\mu}_1(m)\langle\nabla u_m, \nabla\varphi\rangle_{L^2(\Omega)} =
\langle m u_m, \varphi\rangle_{L^2(\Omega)} \quad \forall\, \varphi\in H^1(\Omega) \end{equation*}and
\begin{equation*}\widetilde{\mu}_1(m+ t v)\langle\nabla u_{m+tv},\nabla \varphi\rangle_{L^2(\Omega)} =
\langle (m+tv) u_{m+tv}, \varphi\rangle_{L^2(\Omega)} \quad \forall\, \varphi\in
H^1(\Omega). \end{equation*} By choosing 
$\varphi=u_{m+tv}$ in the former equation, 
$\varphi=u_m$ in the latter and comparing we get
\begin{equation*}
\widetilde{\mu}_1(m+ t v)
\langle m u_m, u_{m+tv}\rangle_{L^2(\Omega)}=
\widetilde{\mu}_1(m)
\langle (m+tv) u_{m+tv}, u_m\rangle_{L^2(\Omega)}.
\end{equation*}Rearranging we find
\begin{equation} \cfrac{\widetilde{\mu}_1(m+ t v)-
\widetilde{\mu}_1(m)}{t}\, \int_\Omega m \, u_m
u_{m +tv}\, dx =\widetilde{\mu}_1(m) \int_\Omega u_m
u_{m +tv} v\, dx .\end{equation} If t goes to zero, then by (ii) of Lemma 3 it follows that 
$u_{m+ t v}$ converges to um in 
$H^1(\Omega)$ and therefore in 
$L^2(\Omega)$. Passing to the limit in (51) and using (29) we conclude
\begin{equation*}
\lim_{t\to 0} \cfrac{\widetilde{\mu}_1(m+ t v)- \widetilde{\mu}_1(m)}{t}\, =
\int_\Omega u_m^2 v\, dx,
\end{equation*}i.e. (50) holds.
Theorem 3. Let 
$m_0\in L_ \lt ^\infty(\Omega)$, 
$\overline{\mathcal{G}(m_0)}$ be the weak* closure in 
$L^\infty(\Omega)$ of the class of rearrangements 
$\mathcal{G}(m_0)$ introduced in Definition 7 and 
$\widetilde{\mu}_1(m)$ defined as in (31) for k = 1. Then
i) there exists a solution of the problem
\begin{equation}
\max_{m\in\overline{\mathcal{G}(m_0)}}\tilde{\mu}_1(m);
\end{equation} (ii) if 
$|\{m_0 \gt 0\}| \gt 0$, any solution 
$\check m_1$ of (52) belongs to 
$\mathcal{G}(m_0)$, more explicitly, we have 
$\tilde\mu_1(m) \lt \tilde\mu_1(\check m_1)$ for all 
$m\in\overline{\mathcal{G}(m_0)}\smallsetminus
\mathcal{G}(m_0)$ (note that, in this case, by Proposition 5 
$\tilde\mu_1(\check m_1)=\mu_1(\check m_1) \gt 0$);
(iii) if 
$|\{m_0 \gt 0\}| \gt 0$, for every solution 
$\check{m}_1\in\mathcal{G}(m_0)$ of (52) there exists an increasing function ψ such that
\begin{equation*}\check{m}_1= \psi(u_{\check{m}_1}) \quad \text{a.e. in }\Omega, \end{equation*} where 
$u_{\check{m}_1}$ is the positive eigenfunction relative to 
$\mu_1(\check{m}_1)$ normalized as in (28).
Proof. i) By (i) of Corollary 3 we have 
$\overline{\mathcal{G}(m_0)}\subset
L^\infty_ \lt (\Omega)$. By (iii) of Proposition 14 and (i) of Lemma 3, 
$\overline{\mathcal{G}(m_0)}$ is sequentially weakly* compact and the map 
$m \mapsto \widetilde{\mu}_1(m)$ is sequentially weakly* continuous respectively. Therefore, there exists 
$\check{m}_1\in
\overline{\mathcal{G}(m_0)}$ such that
\begin{equation*}
\widetilde{\mu}_1(\check{m}_1)=\max_{m\in \overline{\mathcal{G}(m_0)}}\widetilde{\mu}_1(m) .
\end{equation*} (ii) Note that, by Proposition 5, the condition 
$|\{m_0 \gt 0\}| \gt 0$ guarantees 
$\widetilde{\mu}_1(m) \gt 0$ on 
$\mathcal{G}(m_0)$ and then 
$\widetilde{\mu}_1(\check{m}_1) \gt 0$. Let 
$\check{m}_1$ be an arbitrary solution of (52), let us show that 
$\check{m}_1$ actually belongs to 
$\mathcal{G}(m_0)$. Proceeding by contradiction, suppose that 
$\check{m}_1\not \in\mathcal{G}(m_0) $. Then, by (iii) of Proposition 15 and by Definition 8, 
$\check{m}_1$ is not an extreme point of 
$\overline{\mathcal{G}(m_0)}$ and thus there exist 
$m, q \in \overline{\mathcal{G}(m_0)}$ such that m ≠ q and 
$\check{m}_1= \frac{m + q}{2}\, $. By (i) of Lemma 4 and, being 
$\check{m}_1$ a maximizer, we have
\begin{equation*}\widetilde{\mu}_1(\check{m}_1)\leq
\cfrac{\widetilde{\mu}_1(m) +\widetilde{\mu}_1(q)}{2}\,
\leq \widetilde{\mu}_1(\check{m}_1)\end{equation*} and then, equality sign holds. This implies 
$\widetilde{\mu}_1(m) =\widetilde{\mu}_1(q)=\widetilde{\mu}_1(\check{m}_1) \gt 0$, that is m and q are maximizers as well. Now, applying (ii) of Lemma 4 to m and q with 
$t=\frac{1}{2}\, $, we conclude that m and q are linearly dependent and then, by (ii) of Corollary 3, we reach the contradiction m = q. Thus, we conclude that 
$\check{m}_1\in \mathcal{G}(m_0)$ and (ii) is proved. https://epubs.siam.org/doi/book/10.1137/1.9781611971088
(iii) Let 
$\check{m}_1\in \mathcal{G}(m_0)$ be a solution of (52). We prove the claim by using Proposition 17; more precisely, we show that
\begin{equation}
\int_\Omega
\check{m}_1 u_{\check{m}_1}^2\, dx \gt \int_\Omega
m\, u_{\check{m}_1}^2\, dx\end{equation} for every 
$m\in \overline{\mathcal{G}(m_0)}\smallsetminus \{\check{m}_1\}$. By exploiting the convexity of 
$\widetilde{\mu}_1(m)$ (see Lemma 4) and its Gâteaux differentiability in 
$\check{m}_1$ (see Lemma 5) we have (for details see [Reference Ekeland and Témam24])
\begin{equation}
\widetilde{\mu}_1(m)\geq
\widetilde{\mu}_1\big(\check{m}_1)
+\int_\Omega (m-
\check{m}_1) u_{\check{m}_1}^2\, dx
\end{equation} for all 
$m\in \overline{\mathcal{G}(m_0)}$.
First, let us suppose 
$\widetilde{\mu}_1(m) \lt \widetilde{\mu}_1(\check{m}_1)$. Comparing with (54) we find
\begin{equation*}
\int_\Omega
(m-\check{m}_1) u_{\check{m}_1}^2\, dx \lt 0 ,\end{equation*}that is (53).
Next, let us consider the case 
$\widetilde{\mu}_1(m)=\widetilde{\mu}_1(\check{m}_1)$, 
$m\in \overline{\mathcal{G}(m_0)}\smallsetminus \{\check{m}_1\}$. By (ii) there are not maximizers of 
$\widetilde{\mu}_1$ in 
$\overline{\mathcal{G}(m_0)}\smallsetminus \mathcal{G}(m_0)$, therefore 
$m \in \mathcal{G}(m_0)$. Being 
$\check{m}_1\neq m$, by (ii) of Corollary 3, they are linearly independent. Then, (ii) of Lemma 4 implies
\begin{equation*} \widetilde{\mu}_1 \left(\frac{\check{m}_1 + m}{2} \right) \lt \frac{\widetilde{\mu}_1(
\check{m}_1) +\widetilde{\mu}_1(m)}{2}\, = \widetilde{\mu}_1(
\check{m}_1).\end{equation*} Then, arguing as in the previous case with 
$\frac{\check{m}_1 + m}{2}\,$ in place of m we find (53). This completes the proof.
We are now able to prove Theorem 1.
Proof of Theorem 1
Being 
$|\{m_0 \gt 0\}| \gt 0$, we have
\begin{equation*}\lambda_1(m)= \cfrac{1}{\mu_1(m)}\, =\cfrac{1}{\widetilde{\mu}_1(m)}\, \end{equation*} for all 
$m\in\mathcal{G}(m_0)$. Therefore, (i) and (ii) immediately follow by Theorem 3.
(iii) Given that 
$\int_\Omega m_0 \, dx \lt 0$, then, by Proposition 13 and Proposition 15, the negative constant function 
$c=\frac{1}{|\Omega|}\, \int_\Omega m_0 \, dx$ belongs to 
$\overline{\mathcal{G}(m_0)}$. Therefore, by definition of 
$\widetilde{\mu}_1(m)$, 
$\min_{m \in \overline{\mathcal{G}(m_0)}} \widetilde{\mu}_1(m)$ = 0 which, in turns, being 
$\mathcal{G}(m_0)$ dense in 
$\overline{\mathcal{G}(m_0)}$ and 
$\widetilde{\mu}_1(m)$ sequentially weak* continuous, implies 
$\inf_{m \in \mathcal{G}(m_0)}{\mu}_1(m)=0$ and, finally, 
$\sup_{m \in \mathcal{G}(m_0)} \lambda_1(m)=+\infty$.
4. Monotonicity of the minimizers in cylinders
In this section, we consider the optimization problem (3) in cylindrical domains. Here, by (generalized) cylinder we mean a domain of the type 
$\Omega= (0,h) \times \omega\subset \mathbb{R}^N$, where h > 0 and 
$\omega \subset \mathbb{R}^{N-1}$ is a bounded polyhedral or smooth domain. In the sequel, for 
$x\in \mathbb{R}^N$ we will write 
$x=
(x_1, x')$, with 
$x_1\in\mathbb{R}$ and 
$x'= (x_2, \ldots, x_N)\in \mathbb{R}^{N-1}$. Exploiting the notion of monotone (decreasing and increasing) rearrangement, we will able to prove that in a cylinder any minimizer of problem (3) is monotone with respect to x 1. For a comprehensive survey of the monotone rearrangement we use here, we refer the reader to the work of Kawhol (see [Reference Kawhol32]) and Berestycki and Lachand-Robert (see [Reference Berestycki and Lachand-Robert6]). In our paper, for the sake of simplicity, we choose to define this rearrangement only when the domain is a cylinder and, in order to deduce easily some of its properties, as a particular case of the Steiner symmetrization. For a brief summary of the Steiner symmetrization see [Reference Anedda and Cuccu2].
Definition 2. Let 
$\Omega= (0,h) \times \omega$ where h > 0 and 
$\omega \subset \mathbb{R}^{N-1}$ is a bounded polyhedral or smooth domain, and 
$u: \Omega\to \mathbb{R}$ a measurable function bounded from below. Let U be the “x 1-even” extension of u onto 
$(-h,h) \times \omega$ obtained by reflection with respect to the hyperplane 
$\{x\in \mathbb{R}^N: x_1=0\}$ (i.e. 
$U(x_1,x'):=U(-x_1,x')$, 
$x_1\in(-h,0)$, 
$x'\in\omega$) and 
$U^\sharp$ its Steiner symmetrization relative to the same hyperplane. We define the monotone decreasing rearrangement 
$u^\star:\Omega\to\mathbb{R}$ of u to be the restriction of 
$U^\sharp$ on Ω.
In a similar way it can be defined the monotone increasing rearrangement 
$u_\star$. Note that if 
$m\in\mathcal{G}(m_0)$, then 
$m^\star,m_\star\in\mathcal{G}(m_0)$.
From the theory of the Steiner symmetrization (see, for example [Reference Anedda and Cuccu2]) and by Definition 2, we obtain the following first two properties of the monotone decreasing rearrangement.
a) Let
$\Omega= (0,h) \times \omega$, ω as above,
$u: \Omega\to \mathbb{R}$ be a measurable function bounded from below and
$\Phi: \mathbb{R}\to\mathbb{R}$ an increasing function. Then
(55)\begin{equation}
(\Phi(u))^\star=\Phi(u^\star)\quad\text{a.e. in }\Omega.
\end{equation}b) Let
$\Omega= (0,h) \times \omega$, ω as above,
$u, v:\Omega \to
\mathbb{R}$ two measurable functions bounded from below such that
$uv\in L^1(\Omega)$, then the Hardy-Littlewood inequality holds
(56)\begin{equation}
\int_\Omega uv\,dx\leq \int_{\Omega}u^\star v^\star\,dx.
\end{equation}Moreover, from [Reference Berestycki and Lachand-Robert6, Theorem 2.8 and Lemma 2.10] and [Reference Kawhol32, Corollary 2.14] we have
c) Let
$\Omega= (0,h) \times \omega$, ω as above and
$u\in H^1(\Omega)$ a nonnegative function. Then
$u^\star\in H^1(\Omega)$ and the Pòlya-Szegö inequality holds
(57)\begin{equation}
\int_{\Omega}|\nabla u^\star|^2\,dx\leq\int_\Omega|\nabla u|^2\,dx.
\end{equation}
More generally, we have
\begin{equation}
\int_{\Omega}|\nabla_{x'} u^\star|^2\,dx\leq\int_\Omega|\nabla_{x'} u|^2\,dx
\quad\text{and }\quad\int_{\Omega}|(u^\star)_{x_1}|^2\,dx\leq\int_\Omega|u_{x_1}|^2\,dx.
\end{equation}An important ingredient of the next proof is the characterization of the equality case of (57), which is addressed in Theorem 3.1 of [Reference Berestycki and Lachand-Robert6].
We prove Theorem 2.
Proof of Theorem 2
In what follows we use the ideas of the proof of Theorem 2 in [Reference Anedda and Cuccu2]. Let 
$\check m$ be a minimizer of problem (3). By (ii) of Theorem 1, there exists an increasing function ψ such that 
$\check{m}=\psi(u_{\check m})$ a.e. in Ω, where 
$u_{\check{m}}\in V_{\check m}(\Omega)$ denotes the unique positive eigenfunction normalized by 
$\|u_{\check m}\|_{V_{\check m}(\Omega)}= 1$. Therefore, the monotonicity of 
$\check{m}$ is an immediate consequence of the monotonicity of 
$u_{\check m}$. In other words, it suffices to show that either 
$u_{\check m}=u_{\check m}^\star$ or 
$u_{\check m}={(u_{\check m})}_\star$. By using (26) we find
\begin{equation*}
\check\mu_1=\mu_1(\check m)=\cfrac{\int_\Omega \check m u_{\check m}^2
\,dx}{\int_\Omega |\nabla u_{\check m}|^2\,dx}\,.
\end{equation*}The inequality (56), property (55) and Definition 1 yield
\begin{equation*}
\int_\Omega \check m u_{\check m}^2\,dx\leq\int_\Omega \check m^\star (u_{\check m}^\star)^2\,dx=\int_\Omega \check m^\star\big(P_{\check m^\star}(u_{\check m}^\star)
\big)^2\,dx+\frac{1}{\int_\Omega \check m^\star\,dx}\left(\int_\Omega \check m^\star u_{\check m}^\star\,dx\right)^2
\end{equation*}and (57) and (iv) of Proposition 1 give
\begin{equation*}
\int_\Omega |\nabla u_{\check m}|^2\,dx\geq \int_\Omega |\nabla u_{\check m}^\star|
^2\,dx=\int_\Omega |\nabla P_{\check m^\star}(u_{\check m}^\star)|^2\,dx.
\end{equation*} Note that, 
$\check m^\star\in\mathcal{G}(m_0)$ (in particular 
$\int_\Omega \check m^\star\,dx \lt 0$) and 
$P_{\check m^\star}(u_{\check m}^\star)\in V_{{\check m}^\star}$. Exploiting (26) and the maximality of 
$\check\mu_1$, we can write
\begin{equation}
\begin{aligned}
\check\mu_1
&=\cfrac{\int_\Omega \check m u_{\check{m}}^2
\,dx}{\int_\Omega |\nabla u_{\check m}|^2\,dx}
\leq\cfrac{\int_\Omega \check m^\star
(u_{\check m}^\star)^2\,dx}{\int_\Omega |\nabla u_{\check m}^\star|^2\,dx}\\
&\leq\cfrac{\int_\Omega \check m^\star\big(P_{\check m^\star}(u_{\check m}^\star)
\big)^2\,dx+\frac{1}{\int_\Omega \check m^\star\,dx}\left(\int_\Omega \check m^\star u_{\check m}^\star\,dx\right)^2 }{\int_\Omega |\nabla P_{\check m^\star}(u_{\check m}^\star)|^2\,dx}\\
&\leq\cfrac{\int_\Omega \check m^\star\big(P_{\check m^\star}(u_{\check m}^\star)
\big)^2\,dx}{\int_\Omega |\nabla P_{\check m^\star}(u_{\check m}^\star)|^2\,dx}
\leq \cfrac{\int_\Omega \check m^\star (u_{\check m^\star})^2
\,dx}{\int_\Omega |\nabla u_{\check m^\star}|^2\,dx}\,=
\mu_1(\check m^\star)\leq\check\mu_1.
\end{aligned}
\end{equation} Therefore, all the previous inequalities become equalities and yield 
$\int_\Omega\check m^\star
u_{\check m}^\star\,dx=0$, which implies 
$u_{\check m}^\star\in V_{{\check m}^\star}$, and
\begin{equation}
\int_\Omega \check m u_{\check m}^2\,dx=\int_\Omega \check m^\star (u_{\check m}^\star)^2
\,dx,\quad\quad\int_\Omega |\nabla u_{\check m}|^2\,dx= \int_\Omega |\nabla u_{\check m}
^\star|^2\,dx.
\end{equation} Furthermore, by (26), 
$u_{\check{m}}^\star$ is an eigenfunction associated to 
$\mu_1(\check m^\star)$. By the simplicity of 
$\mu_1(\check m^\star)$, 
$u_{\check m}^\star$ being positive in Ω and, by (60), 
$\|u_{\check m}^\star\|_{V_{\check m^\star}(\Omega)}=\|u_{\check m}\|_{V_{\check m}
(\Omega)}=1$, we conclude that 
$u_{\check m}^\star=u_{\check m^\star}$. For simplicity of notation, we put 
$v=u_{\check m}^\star=u_{\check m^\star}$. By (59), 
$\check m^\star$ is a minimizer of (3) and v is the normalized positive eigenfunction of problem (1) associated to 
$1/\mu_1(\check{m}^\star)=\lambda_1(\check{m}^\star)=\check\lambda_1$. Moreover, by (ii) of Theorem 1, there exists an increasing function Ψ such that 
$\check m^\star=\Psi(v)$ a.e. in Ω. Thus v satisfies the problem
\begin{equation}
\begin{cases}-\Delta v =\check\lambda_1 \Psi(v) v\quad &\text{in } \Omega,\\
\frac{\partial v}{\partial \nu}=0 &\text{on } \partial\Omega. \end{cases}
\end{equation} Let 
$C_{0,+}^\infty(\Omega)=\{\varphi\in C_0^\infty(\Omega):\varphi\;\text{is nonnegative}\}$.
From (61) in weak form we have
\begin{equation*}\int_{\Omega} \nabla v\cdot \nabla \varphi_{x_1}\,dx=\check\lambda_1\int_{\Omega}
\Psi(v)v\,\varphi_{x_1}\,dx\quad\forall \varphi\in C^\infty_{0,+}({\Omega}).\end{equation*} Being 
$v\in W^{2,2}_{\rm loc}(\Omega)$ (see [Reference Gilbarg and Trudinger27]), we can rewrite the previous equation as
\begin{equation*}-\int_{\Omega} \nabla v_{x_1}\cdot \nabla \varphi\,dx=\check\lambda_1\int_{\Omega}
\Psi(v)v\,\varphi_{x_1}\,dx.\end{equation*} Adding 
$\check\lambda_1\int_{\Omega}
\Psi(v)v_{x_1}\,\varphi\,dx$ to both sides and since 
$v\in C^{1,\beta}(\Omega)$ for all 
$\beta\in(0,1)$ (see [Reference Gilbarg and Trudinger27]), it becomes
\begin{equation}
-\int_{\Omega} \nabla v_{x_1}\cdot \nabla \varphi\,dx+\check\lambda_1\int_{\Omega}
\Psi(v)v_{x_1}\,\varphi\,dx=\check\lambda_1\int_{\Omega} \Psi(v)(v\,\varphi)_{x_1}\,dx.
\end{equation} Let us show that 
$\int_{\Omega} \Psi(v)(v\,\varphi)_{x_1}\,dx\geq 0.$ By Fubini’s Theorem we get
\begin{equation}
\int_{\Omega} \Psi(v)(v\,\varphi)_{x_1}\,dx=
\int_{\omega}dx'\int_0^h \Psi(v)(v\,\varphi)_{x_1}\,dx_1.
\end{equation} For any fixed 
$x'\in\omega$, let 
$\alpha=\alpha(x_1)=v(x_1,x')\,\varphi(x_1,x')$. Since φ has compact support, we can consider α trivially defined on the whole 
$[0, h]$. Since 
$\alpha(x_1)$ is continuous and 
$\Psi(v)$ is decreasing with respect to x 1, the Riemann-Stieltjes integral 
$\int_0^{h} \Psi(v)\,d\alpha(x_1)$ is well defined (see Theorem 7.27 and the subsequent note in [Reference Apostol4]). Moreover, by using [Reference Apostol4, Theorem 7.8] we have
\begin{equation}\quad
\int_0^{h} \Psi(v)(v\,\varphi)_{x_1}\,dx_1=\int_0^{h} \Psi(v)\,d\alpha(x_1).
\end{equation} By [Reference Apostol4, Theorems 7.31 and 7.8] there exists a point x 0 in 
$[0, h]$ such that
\begin{equation*}
\begin{aligned}
-\int_0^{h}\Psi(v)\,d\alpha(x_1)&=-\Psi(v(0,x'))\int_0^{x_0} \,d\alpha(x_1)-\Psi(v(h, x'))\int_{x_0}^{h} \,d\alpha(x_1)\\
&=-\Psi(v(0,x'))\int_0^{x_0}(v\,\varphi)_{x_1}\,dx_1-\Psi(v(h,x'))\int_{x_0}^{h}(v\,\varphi)_{x_1}\, dx_1.\\
\end{aligned}
\end{equation*} Computing the integrals and recalling that 
$\varphi\in C^\infty_{0,+}({\Omega})$, v is positive and 
$\Psi(v)$ is decreasing with respect to x 1, we conclude that
\begin{equation*}-\int_0^{h}\Psi(v)\,d\alpha(x_1)=v(x_0,x')\varphi(x_0,x')\left[\Psi(v(h,x'))-\Psi(v(0,x'))\right]\leq 0.\end{equation*} Therefore, by the previous inequality and (64) it follows 
$\int_0^{h} \Psi(v)(v\,\varphi)_{x_1}\,dx_1\geq 0$ for any 
$x'\in\omega$ and, in turn, from (63) we obtain 
$\int_{\Omega} \Psi(v)(v\,\varphi)_{x_1}\,dx\geq 0$. Hence, by (62), 
$v_{x_1}$ satisfies the differential inequality
\begin{equation*}
\Delta v_{x_1}+\check\lambda_1\Psi(v)v_{x_1}\geq 0\quad \text{in }\Omega
\end{equation*} in weak form. Then, applying [57, Theorem 2.5.3] and since 
$v_{x_1}\leq 0$ in Ω, we conclude that either 
$v_{x_1}\equiv 0$ or 
$v_{x_1} \lt 0$.
In the first case, v, and then 
$u_{\check m}$, is constant with respect to x 1.
Let 
$v_{x_1} \lt 0$. By the second equality of (60) and (58) we obtain
\begin{equation*}
\int_{\Omega}|(u_{\check m})_{x_1}|^2\,dx=\int_\Omega|{(u_{\check m}^\star)}_{x_1}|^2\,dx.
\end{equation*}By Fubini’s Theorem it becomes
\begin{equation}
\int_{\omega}dx'\int_0^h |(u_{\check m})_{x_1}|^2\,dx_1=
\int_{\omega}dx'\int_0^h |{(u_{\check m}^\star)}_{x_1}|^2\,dx_1.
\end{equation} Being 
$u_{\check m}$ and 
$u_{\check m}^\star=u_{\check m^\star}$ of class 
$C^{1,\beta}(\Omega)$, the functions 
$x'\mapsto\int_0^h |(u_{\check m})_{x_1}|^2\,dx_1$ and 
$x'\mapsto\int_0^h |{(u_{\check m}^\star)}_{x_1}|^2\,dx_1$ are continuous on ω. Therefore, using identity (65) and (57) in the one dimensional case we find
\begin{equation*}
\int_0^h |(u_{\check m})_{x_1}|^2\,dx_1=
\int_0^h |{(u_{\check m}^\star)}_{x_1}|^2\,dx_1\quad\forall x'\in\omega.
\end{equation*} From Theorem 3.1 in [Reference Berestycki and Lachand-Robert6], again in the one dimensional case, we conclude that for all 
$x'\in \omega$ either 
$u_{\check m}=u_{\check m}^\star $ or 
$u_{\check m}
=(u_{\check m})_\star$. This implies that for all 
$x'\in \omega$ either 
$(u_{\check m})_{x_1}
=(u_{\check m}^\star)_{x_1}=v_{x_1} \lt 0$ or 
$(u_{\check m})_{x_1}=((u_{\check
m})_\star)_{x_1}=-(u_{\check m}^\star)_{x_1}=-v_{x_1} \gt 0$. Being 
$(u_{\check m})_{x_1}$ continuous in Ω and Ω an open connected set, it follows that 
$(u_{\check
m})_{x_1}$ does not change sign in Ω. Equivalently either 
$u_{\check m}=u_{\check
m}^\star $ or 
$u_{\check m}=(u_{\check m})_\star$ in the whole Ω. Finally, by 
$\check{m}=\psi(u_{\check m})$, we conclude that either 
$\check m=\check m^\star$ or 
$\check m=\check m_\star$. This proves the theorem.
Remark 3. Similar results on the monotonicity of the minimizers can be found in [Reference Jha and Porru30, Theorem 2.4] in the one dimensional case and for an arbitrary m 0 and in [Reference Lamboley, Laurain, Nadin and Privat33, Proposition 5] in general dimension, for an orthotope and m 0 of “bang-bang” type. Both the previous results can be obtained from Theorem 2. We also mention that other qualitative features of the minimizers are known. We refer the reader again to [Reference Jha and Porru30], where a symmetry breaking result in dimension two when Ω is an annulus is given, and to [Reference Lamboley, Laurain, Nadin and Privat33] for further qualitative properties of the minimizers in the case of the orthotope and the ball. In particular, in this last case it is proved that a minimizer cannot be a ball concentric to Ω. Finally, it is worth noting that the monotonicity property stated in Theorem 2 has also been numerically observed by some authors (see [Reference Kao, Lou and Yanagida31, Reference Lamboley, Laurain, Nadin and Privat33, Reference Roques and Hamel42]).
Acknowledgements
The authors would like to thank the anonymous referee for his suggestions and comments.
Funding statement
The authors are partially supported by the research project Analysis of PDEs in connection with real phenomena, CUP F73C22001130007, funded by Fondazione di Sardegna, annuity 2021. The authors are members of GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica “Francesco Severi”).
The authors also acknowledge the financial support under the National Recovery and Resilience Plan (NRRP), Mission 4 Component 2 Investment 1.5 - Call for tender No.3277 published on December 30, 2021 by the Italian Ministry of University and Research (MUR) funded by the European Union—NextGenerationEU. Project Code ECS0000038—Project Title eINS Ecosystem of Innovation for Next Generation Sardinia—CUP F53C22000430001- Grant Assignment Decree No. 1056 adopted on June 23, 2022 by the Italian Ministry of University and Research (MUR).
Appendix A. Rearrangements of measurable functions
In this appendix we introduce the concept of rearrangement of a measurable function and summarize some related results we use in the previous sections. The idea of rearranging a function dates back to the book [Reference Hardy, Littlewood and Pólya28] of Hardy, Littlewood and Pólya, since than many authors have investigated both extensions and applications of this notion. Here we relies on the results in [Reference Alvino, Trombetti and Lions1, Reference Burton10, Reference Burton11, Reference Day20, Reference Kawhol32, Reference Ryff43].
Let Ω be an open bounded set of 
$\mathbb{R}^N$.
Definition 3. For every measurable function 
$f:\Omega\to\mathbb{R}$ the function 
$d_f:\mathbb{R}\to [0,|\Omega|]$ defined by
\begin{equation*}d_f(t)=|\{x\in\Omega: f(x) \gt t\}|\end{equation*}is called distribution function of f.
The symbol µf is also used. It is easy to prove the following properties of df.
Proposition 9. For each f the distribution function df is decreasing, right continuous and the following identities hold true
\begin{equation*}\lim_{t\to-\infty}d_f(t)=|\Omega|,\quad\quad \lim_{t\to\infty}d_f(t)=0.\end{equation*}Definition 4. Two measurable functions 
$f,g:\Omega \to \mathbb{R}$ are called equimeasurable functions or rearrengements of one another if one of the following equivalent conditions is satisfied
(i) 
$|\{x\in \Omega: f(x) \gt t\}|=|\{x\in \Omega: g(x) \gt t\}| \quad \forall\, t\in \mathbb{R}$;
(ii) 
$d_f=d_g$.
Equimeasurability of f and g is denoted by 
$f\sim g$. Equimeasurable functions share global extrema and integrals as it is stated precisely by the following proposition.
Proposition 10. Suppose 
$f\sim g$ and let 
$F:\mathbb{R}\to\mathbb{R}$ be a Borel measurable function, then
i)
$|f|\sim |g|$;ii)
${\mathrm{ess}\sup\,} f={\mathrm{ess}\sup\,} g$ and
${\mathrm{ess}\inf\,} f={\mathrm{ess}\inf\,} g$;iii)
$F\circ f\sim F\circ g$;iv)
$F\circ f\in L^1(\Omega)$ implies
$F\circ g\in L^1(\Omega)$ and
$\int_\Omega F\circ f\,dx=
\int_\Omega F\circ g\,dx$.
For a proof see, for example, [Reference Day20, Proposition 3.3] or [Reference Burton11, Lemma 2.1].
In particular, for each 
$1\leq p\leq\infty$, if 
$f\in L^p(\Omega)$ and 
$f\sim g$ then 
$g\in L^p(\Omega)$ and
\begin{equation*}
\|f\|_{L^p(\Omega)}=\|g\|_{L^p(\Omega)}.
\end{equation*}Definition 5. For every measurable function 
$f:\Omega\to\mathbb{R}$ the function 
$f^*:(0,|\Omega|)\to
\mathbb{R}$ defined by
\begin{equation*}f^*(s)=\sup\{t\in\mathbb{R}: d_f(t) \gt s\}\end{equation*}is called decreasing rearrangement of f.
An equivalent definition (used by some authors) is 
$f^*(s)=\inf\{t\in\mathbb{R}: d_f(t)\leq
s\}$.
Proposition 11. For each f its decreasing rearrangement 
$f^*$ is decreasing, right continuous and we have
\begin{equation*}\lim_{s\to 0}f^*(s)={\mathrm{ess}\sup\,} f\quad\text{and}\quad\lim_{s\to|\Omega|}f^*(s)={\mathrm{ess}\inf\,} f.\end{equation*} Moreover, if 
$F:\mathbb{R}\to\mathbb{R}$ is a Borel measurable function then 
$F\circ f\in L^1(\Omega)$ implies 
$F\circ f^*\in L^1(0,|\Omega|)$ and
\begin{equation*}\int_\Omega F\circ f\,dx=\int_0^{|\Omega|} F\circ f^*\,ds.\end{equation*} Finally, 
$d_{f^*}=d_f$ and, for each measurable function g we have 
$f\sim g$ if and only if 
$f^*=g^*$.
Some of the previous claims are simple consequences of the definition of 
$f^*$, for more details see [Reference Day20, Chapter 2].
As before, it follows that, for each 
$1\leq p\leq\infty$, if 
$f\in L^p(\Omega)$ then 
$f^*\in L^p(0,|\Omega|)$ and 
$\|f\|_{L^p(\Omega)}=\|f^*\|_{L^p(0,|\Omega|)}.$
Definition 6. Given two functions 
$f,g\in L^1(\Omega)$, we write 
$g\prec f$ if
\begin{equation*}\int_0^t g^*\,ds\leq \int_0^t f^*\,ds\quad\forall\; 0\leq t\leq|\Omega|\quad\quad
{and}\quad\quad\int_0^{|\Omega|} g^*\,ds= \int_0^{|\Omega|} f^*\,ds.\end{equation*} Note that 
$g\sim f$ if and only if 
$g\prec f$ and 
$f\prec g$. Among many properties of the relation 
$\prec$ we mention the following (a proof is in [Reference Day20, Lemma 8.2]).
Proposition 12. For any pair of functions 
$f,g\in L^1(\Omega)$ and real numbers α and β, if 
$\alpha\leq f\leq\beta$ a.e. in Ω and 
$g\prec f$ then 
$\alpha\leq g\leq\beta$ a.e. in Ω.
Proposition 13. For 
$f\in L^1(\Omega)$ let 
$g=\frac{1}{|\Omega|}\int_\Omega f \, dx$. Then we have 
$g\prec f$.
Definition 7. Let 
$f:\Omega\to\mathbb{R}$ a measurable function. We call the set
\begin{equation*}\mathcal G(f)=\{g:\Omega\rightarrow\mathbb{R}:g\ \text{is measurable and}\ g\sim f\}\end{equation*}the class of rearrangements of f or the set of rearrangements of f.
Note that, for 
$1\leq p\leq\infty$, if f is in 
$L^p(\Omega)$ then 
$\mathcal{G}(f)$ is contained in 
$L^p(\Omega)$.
In this paper we are interested in studying the optimization of a functional which is defined on a class of rearrangements 
$\mathcal{G}(m_0)$, where m 0 belongs to 
$L^\infty(\Omega)$. For this reason, although almost all of what follows holds in a much more general context, hereafter we restrict our attention to classes of rearrangements of functions in 
$L^\infty(\Omega)$. We need compactness properties of the set 
$\mathcal{G}(m_0)$, with a little effort it can be shown that this set is closed but in general it is not compact in the norm topology of 
$L^\infty(\Omega)$. Therefore we focus our attention on the weak* compactness. By 
$\overline{\mathcal{G}(m_0)}$ we denote the closure of 
$\mathcal{G}(m_0)$ in the weak* topology of 
$L^\infty(\Omega)$.
Proposition 14. Let m 0 be a function of 
$L^\infty(\Omega)$. Then 
$\overline{\mathcal{G}(m_0)}$ is
i) weakly* compact;
ii) metrizable in the weak* topology;
iii) sequentially weakly* compact.
For the proof see [Reference Anedda, Cuccu and Frassu3, Proposition 3.6].
Moreover, the sets 
$\mathcal{G}(m_0)$ and 
$\overline{\mathcal{G}(m_0)}$ have further properties.
Definition 8. Let C be a convex set of a real vector space. An element v in C is said to be an extreme point of C if for every u and w in C the identity 
$v=\frac{u+w}{2}$ implies u = w.
A vertex of a convex polygon is an example of extreme point.
Proposition 15. Let m 0 be a function of 
$L^\infty(\Omega)$, then
i)
$\overline{\mathcal{G}(m_0)}=\{f\in L^\infty(\Omega): f\prec m_0\}$,ii)
$\overline{\mathcal{G}(m_0)}$ is convex,iii)
$\mathcal{G}(m_0)$ is the set of the extreme points of
$\overline{\mathcal{G}(m_0)}$.
Proof. The claims follow from [Reference Day20, Theorems 22.13, 22.2, 17.4, 20.3].
An evident consequence of the previous theorem is that 
$\overline{\mathcal{G}(m_0)}$ is the weakly* closed convex hull of 
$\mathcal{G}(m_0)$.
Corollary 3. Let 
$m_0\in L^\infty(\Omega)$ and 
$m,q\in\overline{\mathcal{G}(m_0)}$. Then
i)
$\int_\Omega m\,dx=\int_\Omega m_0\,dx$;ii) assuming
$\int_\Omega m_0\,dx\neq 0$, m = q if and only if m and q are linearly dependent.
Proof. (i) It follows immediately by (i) of Proposition 15, Definition 6 and Proposition 11 with F equal to the identity function.
(ii) If m and q are linearly dependent, then, without loss of generality we can assume that 
$m=\alpha q$, for some 
$\alpha\in\mathbb{R}$. Integrating over Ω and using (i) we find m = q.
The following is [Reference Day20, Theorem 11.1] rephrased for our case.
Proposition 16. Let 
$u\in L^1(\Omega)$ and 
$m_0\in L^\infty(\Omega)$. Then
\begin{equation}
\int_0^{|\Omega|} m_0^*(|\Omega|-s) u^*(s)\, ds\leq\int_\Omega m\,u\, dx
\leq\int_0^{|\Omega|} m_0^*(s) u^*(s)\, ds\quad\quad\forall m\in\mathcal{G}(m_0),
\end{equation}and moreover both sides of (66) are achieved.
The previous proposition implies that the linear optimization problems
\begin{equation}
\sup_{m \in \mathcal{G}(m_0)} \int_\Omega m u \, dx
\end{equation}and
\begin{equation*}
\inf_{m \in \mathcal{G}(m_0)} \int_\Omega m u \, dx
\end{equation*}admit solution.
Finally, we recall the following result proved in [Reference Burton10, Theorem 5].
Proposition 17. Let 
$u\in L^1(\Omega)$ and 
$m_0\in L^\infty(\Omega)$. If problem (67) has a unique solution mM, then there exists an increasing function ψ such that 
$m_M=\psi \circ u$ a.e. in Ω.
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