Optimisation of the total population size for logistic diffusive equations: bang-bang property and fragmentation rate

In this article, we give an in-depth analysis of the problem of optimising the total population size for a standard logistic-diffusive model. This optimisation problem stems from the study of spatial ecology and amounts to the following question: assuming a species evolves in a domain, what is the best way to spread resources in order to ensure a maximal population size at equilibrium? {In recent years, many authors contributed to this topic.} We settle here the proof of two fundamental properties of optimisers: the bang-bang one which had so far only been proved under several strong assumptions, and the other one is the fragmentation of maximisers. Here, we prove the bang-bang property in all generality using a new spectral method. The technique introduced to demonstrate the bang-bang character of optimizers can be adapted and generalized to many optimization problems with other classes of bilinear optimal control problems where the state equation is semilinear and elliptic. We comment on it in a conclusion section.Regarding the geometry of maximisers, we exhibit a blow-up rate for the $BV$-norm of maximisers as the diffusivity gets smaller: if $\O$ is an orthotope and if $m_\mu$ is an optimal control, then $\Vert m_\mu\Vert_{BV}\gtrsim \sqrt{\mu}$. The proof of this results relies on a very fine energy argument.


Introduction
This article is devoted to the study of a problem of calculus of variations motivated by questions of spatial ecology. This problem is related to the ubiquitous question of optimal location of resources. While we further specify what we mean by "optimal" in what follows, let us note that optimisation problems related to the location of resources are a possible way to tackle the question of spatial heterogeneity in reaction-diffusion equations. In this context, spatial heterogeneity is interpreted as heterogeneity of the resources available to a given population.
In this paper, we thoroughly analyse the issue of optimising the total population size with respect to the resource distribution. The reaction-diffusion model we deal with is made precise in Section 1.1 and the precise statement of our main results in Section 1.2. In a nutshell, our results may be recast as follows: • First, we give a characterisation of pointwise properties of optimal resource distributions (also called the bang-bang property) that has been partially tackled in [21,25]; in these previous contributions, the contents of which we discuss in Sections 1.1 and 1.3, partial answers are provided under several technical assumptions. We present here a new method that we believe to be flexible and versatile enough to be applied to a wide class of bilinear optimal control problem, and that provides a positive answer to the question of knowing whether optimal resource distributions are bang-bang.
• Second, we prove a fragmentation phenomenon, with explicit blow-up rates: as has been noticed [24,21,22], for the optimisation of the total population size, the characteristic dispersal rate of the population has a drastic influence on the geometry of optimal resource distributions (in the sense that, the lower the characteristic dispersal rate, the more spread out the optimal resource distribution). Here, we provide an explicit blow-up rates for the BV -norm of optimal resource distribution, the BV -norm being a natural way to quantify the fragmentation or complexity of a resource distribution. We refer to Section 1.1 for further explanations.

Model and statement of the problems
Statement of the problems Let us first lay down the model and the optimisation problems under consideration. The following paragraph is dedicated to explaining which kind of properties we want to obtain for these optimisation problems. We introduce the model we consider throughout the paper. We place ourselves in the framework of the Fisher-KPP equation which, since the seminal [8,11], has been used at length: while its apparent simplicity makes it amenable to mathematical analysis, it is complex enough to capture several fundamental aspects of population dynamics [27]. This model reads: where θ : Ω → IR + is the population density,. The population access to resources is modelled by a function m ∈ L ∞ (Ω) and µ > 0 is the dispersal rate.
Although we consider here Neumann boundary conditions, Theorems I and II below can be extended to Robin boundary conditions as well, the only difficulty being that one would need to ensure existence and uniqueness for the logistic-diffusive equations under these conditions. We comment on this in the conclusion (Section 4).
We can hence define the total-population size functional ∀µ > 0, ∀m ∈ M(Ω), F µ (m) :=ˆΩ θ m,µ . (1.1) We use the following class of constraints on the admissible resource distributions m, which was introduced in [17] and used, for instance, in [21,25]: The parameter m 0 is a positive real number such that m 0 < |Ω|, where |Ω| denotes the volume of Ω, in order to ensure that M(Ω) = ∅. The L 1 constraint accounts for the fact that, in a given domain, only a limited amount of resources is available. The second constraint is a pointwise one, and accounts for natural limitations of the environment, i.e. the fact that, in a single spot, only a maximum amount of resources can be available. The optimisation problem we consider reads sup m∈M(Ω) where F µ (m) is given by (1.1).

Remark 1 (Existence of maximisers).
For any µ > 0, the existence of a solution m * µ of (P µ ) is an immediate consequence of the direct method in the calculus of variations.
In the following paragraph, we present the fundamental properties we are interested in.
Optimisation of spatial heterogeneity in mathematical biology: fundamental properties under consideration Starting from spatially homogeneous models [8,11], in which a population is assumed to live in a homogeneous environment, mathematical biology has over the past decades started considering the impact of spatial heterogeneity on population dynamics [5]. In most works, this spatial heterogeneity is modelled using resource distributions. Mathematically, this amounts to taking into account the heterogeneity in the reaction term of the equation. Given that it is hopeless, for a given resource distribution, to attain an explicit description of the ensuing population dynamics, the focus has, more recently, shifted to an optimisation point of view.
This approach has been initiated in [3,10,16] and has since received a considerable amount of attention [2,6,13,24,19,21,25]. The initial question that motivated most of these works was related to the optimal survival ability of a population [3,26]. Namely: What is the best way to spread resources in a domain to ensure the optimal survival of a population?
This problem is by now very well understood in several simple cases (we provide ampler references in Section 1.3). Among all the issues tackled by the authors of [3,10,13], let us single out the following ones, which have been deemed crucial in the study of spatial heterogeneity as they provide simple, qualitative information about the influence of heterogeneity: in a domain Ω, if we consider resource distribution m belonging to M(Ω) defined by (1.2), 1. does the bang-bang property hold at the optimum? In other words, if one looks at maximising a criterion over resource terms in M(Ω), does any optimal resource distribution m * write m * = 1 E , for some measurable subset E of Ω of positive measure? Alternatively, this means that the underlying domain Ω can be decomposed as Despite several partial results [21,25] which we detail in Remarks 4 and 5 , this property is not known to hold in general for the optimisation of the total population size. In this article we prove that this property indeed holds for the optimal population size whatever the value of µ > 0 be (Theorem I).
2. do optimal resources tend to concentrate? In "simple "cases (i.e. in simple geometries and for specific boundary conditions), optimal resource distributions for the survival ability [3,10] are known to be concentrated. For instance, considering an optimal resource distribution for the survival ability, which is known to write m * = 1 E , then the set E is connected and enjoys moreover a symmetry property for Neumann boundary conditions in an orthotope [3,Proposition 2.9]. A similar conclusion holds whenever Ω = B(0; r) is a ball and if Dirichlet boundary conditions are imposed rather than Neumann. In that case, E = B(0; r * ) is another centered ball, with a radius r * chosen so as to satisfy the volume constraint. For general geometries and Robin boundary conditions, the situation is very involved and we refer to [13] for up to date qualitative properties. Such results are a mathematical formalisation of a paradigm first stated in [26]: fragmenting the set {m = 1} leaves less chance for survival: concentrating resources is favorable to population dynamics.
In the case of the total population size, it was first noticed in [21] that such results do not in general hold for small diffusivities, where the geometry of the optimal resource distribution tends to become more complicated. Recently, in [24], a complete treatment of a spatially discretised version of the problem was carried out, and precise fragmentation rules were established. However, these results cannot be extended to the present continuous version, since the optimiser they compute strongly depends on the discretization scale. In [22], it was shown that, the slower the dispersal rate of the population, the bigger the BV -norm 1 of the optimal resource distribution is.
Remark 2. When m ∈ W 1,1 (Ω), the BV -norm and the W 1,1 norm coincide. When m = 1 E and m is a Cacciopoli set (i.e. a set with finite Cacciopoli perimeter) then m BV (Ω) = |E| + Per(E), where Per(E) is the Cacciopoli perimeter of the set. As a consequence, in our context, an information on the blow-up rate of the BV -norm yields an information on the blow-up rate of the T V -norm and, since Theorem I ensures that any maximiser m * µ writes as 1 E * µ , this implies a blow-up rate on Per(E * µ ) as µ → 0 + . We refer to [1] for more information regarding functions of bounded variations and perimeters of sets.
In [22], the main result reads: In this article, we quantify this result by explicitly identifying blow-up rates in terms of the characteristic dispersal rate, and provide a scaling we expect to be optimal (Theorem III). The proof relies on fine energy estimates.
A more in-depth discussion of the bibliography is included in Section 1.3. 1 Recall that the total variation semi-norm of a function is and that the bounded variation norm of m is in turn defined as (1.5)

The bang-bang property
Let us first state that every solution of the optimal population size problem is bang-bang. This property, intrinsically interesting, has a practical interest: it allows us to reformulate the problem as a shape optimization one, the unknown being the set in which m takes its maximum value. One can then use adapted numerical approaches.
Theorem I. Let Ω ⊂ IR d be a bounded connected domain with a C 2 boundary. Let m * µ be a solution of (P µ ). Then there exists a measurable subset E ⊂ Ω such that Remark 3 (Sketch of the proof). The idea of the proof rests upon the following fact: we can actually show that the second order Gâteaux derivative of the criterion F µ at a point m ∈ M(Ω) in a direction h (such that m + th ∈ M(Ω) for t small enough) writes where L m denotes an elliptic operator of second order. We then argue by contradiction, assuming the existence of a maximiser m * µ that is not a bang-bang function, meaning that the set {0 < m * µ < 1} is of positive Lebesgue measure. Using the expression ofF µ (m)[h, h] above, we exhibit a function h in L ∞ supported in {0 < m * µ < 1}, with´Ω h = 0, such that´Ω |∇θ m * µ ,µ | 2 is much larger than´Ωθ 2 m * µ ,µ . This is done by using the Fourier (spectral) expansion of θ m,µ , associated with the operator L m * µ , and by choosing h as above, and such that, hθ m * µ ,µ only has high Fourier modes in this basis.
Remark 4 (Comparison with the results of [25]). In [25], the following result is proved: if m ∈ M(Ω) is such that {0 < m < 1} has a non-empty interior, then it is not a solution of (P µ ). This in particular implies that, if a maximiser m * µ of the total population size functional is Riemann integrable, then m * µ is continuous almost everywhere in Ω and is thus necessarily of bang-bang type. However, such regularity is usually extremely hard to prove, and it is unclear to us whether it is attainable in this context. Furthermore, we believe we have located a slight mistake in their proof and we hence provide a correction in Section 2.2, where we also comment on the comparison between our two proofs.
Remark 5. In [21], the bang-bang property is proved to hold whenever the diffusivity µ is large enough, using a proof that is also based on a second derivative calculation, but whose philosophy is completely different from that of Theorem I. Our present result does not require such an assumption. Remark 6. A minor adaptation of our proof allows us to handle more general admissible sets and criteria: • let us consider a function j satisfying j ∈ C 2 ([0; 1]; IR), j is increasing in [0; 1]: j ′ > 0.
(H j ) We define, for any µ > 0, and the optimisation problem sup Then proving a bang-bang property for this problem is amenable to analysis using our technique.
• If one were to change the L ∞ bounds on m to 0 m κ for some positive κ, the only modification would be to replace [0; 1] with the interval [0; κ] in assumption (H j ) above.
We claim that our method of proof fits immediately to show the following result: Theorem II. Let Ω ⊂ IR d be a C 2 bounded domain and let j satisfying (H j ). Let m * µ,j be a solution of (P j,µ ). Then m * µ,j is a bang-bang function: there exists a measurable subset A short paragraph explaining how to adapt the proof of Theorem I is provided in Section 2.3.

Quantifying the fragmentation for small diffusivities
Our second main result deals with the aforementioned fragmentation property for low diffusivities.
Hence, according to [22,Lemma 2], one has The equality on the right-hand side is obtained in [16, Theorem 1.2].
Remark 7. If we consider another domainΩ such that (1.10) holds, then the main result below, Theorem III, holds inΩ.
We provide hereafter an explicit blow-up rate that we believe to be optimal. Once again, let us emphasize that it this rate does not depend on the space dimension d.
Theorem III. Let d 1 and let Ω = (0; 1) d . There exists C 0 > 0 such that the following holds: there exists µ 0 > 0 such that, for any µ ∈ (0; µ 0 ), if m * µ is a solution of (P µ ), then Remark 8 (Comment on the proof of Theorem III). The crux of the proof is the variational formulation of (E m,µ ), which ensures that θ m,µ is the unique minimiser of 12) and which needs to be carefully estimated as µ → 0 + . We prove that a "shifted" version of this energy controls the quantity θ m,µ − m L 1 (Ω) (Lemma 18). Therefore, using estimate (1.10), we aim at controlling E m,µ (θ m,µ ) as µ → 0 + . Using Modica-type estimates, one can show that, for a fixed m ∈ M(Ω) that writes However, this convergence is non-uniform with respect to m (or, more precisely to E) and, since we are working with a maximisation problem, it is not possible to conclude using the convergence result above. In the one dimensional case, we propose, in the appendix, an adaptation of [23] that makes this strategy work nonetheless. In higher dimension, we estimate the energy using a regularisation of m as a test function in the energy formulation of the equation.

Bibliographical comments on (P µ )
In this section, we gather a discussion on references connected to the optimisation of the total population size in logistic-diffusive models. For a presentation of the literature devoted to the optimal survival ability, we refer to [18,Introduction].
Influence of the diffusivity µ on F µ . Problem (P µ ) has been first introduced in [17] and several properties had been derived in [16], one of which is the following: for every µ > 0, the This result means that spatial homogeneity is detrimental to the population size. Furthermore, it is proved in [16] that, if m ∈ M(Ω) is given, then (1.14) Hence, for a given m ∈ M(Ω), the low and high diffusivity limits of the functional correspond to global minima. However, it was proved in [22, Lemma 2] that lim inf showing the intrinsic difficulty of passing to the low-diffusivity limit in problem (P µ ). This point of view, where the resource distribution is considered fixed and the diffusivity is taken as a variable, was later deeply analysed in several articles. Notable among these are the following results: 1. In [2], for a fixed m ∈ L ∞ (Ω) such that m(·) 0 and m(·) = 0, the authors consider the optimisation problem and observe that, in the one-dimensional case Ω = (0; 1), there holds This bound is sharp (a maximising sequence is explicitly constructed) and is not reached by any function m. This work has been later extended to the higher-dimensional case in [9] and the authors prove that, in that case (i.e. in dimension d 2), there holds In [14], a function m such that the map µ → F µ (m) has several local maxima is constructed. It emphasizes the intrinsic complexity of the interplay between the population size functional and the parameter µ > 0.
Finally, let us also note that a related problem, where the underlying model is a system of ODEs with identical migration rates, was considered in [15].
We also point out to two surveys [12,20] and to the references therein for up-to-date considerations about the influence of spatial heterogeneity for single or multiple species models or for optimisation problems in mathematical biology.
2 Proofs of Theorems I and II

Proof of Theorem I
The proof of this Theorem relies on a new formulation of the second order optimality conditions for the problem (P µ ). Let us first compute the necessary optimality conditions of the first and second orders.
Computation of optimality conditions It is established in [6, Lemma 4.1] that, for any µ > 0 the map M(Ω) ∋ m → θ m,µ is differentiable at the first order in the sense of Gâteaux. Adapting their proof yields without difficulty its second order Gâteaux-differentiability. Let us fix m ∈ M(Ω) and an admissible perturbation 2 h ∈ L ∞ (Ω). Let us denote byθ m,µ (resp.θ m,µ ) the first (resp. second) Gâteaux-derivative of θ ·,µ at m in the direction h. It is standard (we refer to [6,Lemma 4.1]) to see thatθ m,µ solves Remark 9. The fact thatθ m,µ is uniquely determined by that equation (in other words, that (2.1) has a unique solution can be proved as in [6,21]. For the sensitivity analysis and computation of the Gâteaux-derivatives, we also refer to [25].
To derive a tractable equation for the Gâteaux derivativeḞ µ (m)[h] of the functional F µ at m in the direction h, let us introduce the adjoint state p m,µ as the solution of so that, multiplying (2.1) by p m,µ and integrating by parts readily giveŝ It is standard in optimal control theory (see e.g. [28]) that, if m * µ is a solution of (P µ ) then there exists a constant c such that Remark 10. As is done in [21], the sets {m * µ = 1} and {m * µ = 0} can be described in terms of level sets of the so-called switching function θ m,µ p m,µ but we do not detail it since these are not informations we will use in the proof.
Let us turn to the computation of the second order Gâteaux derivative of the functional F µ in the direction h, which will be denotedF µ (m) [h, h]. To obtain it, we first recall (see [21,Equation (18) Multiplying (2.5) by p m,µ and integrating by parts yieldŝ Let us introduce u m,µ := pm,µ θm,µ . We thus obtain Furthermore, we have the following result: Proof of Lemma 11. We start from the observation that θ m,µ solves (E m,µ ) implies that the principal eigenvalue λ(m − θ m,µ , µ) of the operator −µ∆ − (m − θ) Id is zero [16]. Since θ m,µ > 0 in Ω, the first eigenvalue λ(m − 2θ m,µ , µ) of the operator L m : as a consequence of the monotonicity of eigenvalues [7]. Since p m,µ satisfies L m p m,µ = 1 > 0 with Neumann boundary conditions, the conclusion follows from multiplying the equation on p m,µ by the negative part (p m,µ ) − and integrating by parts: it yields However, according to the Courant-Fischer principle, and therefore, it follows that p m,µ (·) 0 and p m,µ (·) = 0 in Ω. To conclude, it suffices to apply the strong maximum principle.
If we then define V m,µ := m − θ m,µ + , which leads to a contradiction whenever ε > 0 is chosen small enough. It is standard to show that any perturbation h supported in Ω is admissible if, and only if´Ω h = 0.

Remark 12.
To implement the previous construction, it suffices in fact to construct h ∈ L 2 (Ω) so that (2.17) is satisfied and´Ω h = 0, forgetting that h has to belong to L ∞ (Ω). Indeed, let us assume that such a h ∈ L 2 (Ω) exists. Then, we introduce the sequence h n := h1 |h| n −´Ω h1 |h| n ∈ L ∞ (Ω), which converges weakly in L 2 (Ω) to h as n → ∞. By elliptic regularity, it entails strong W 1,2regularity ofθ m,µ [h n ] toθ m,µ as n → ∞ and thus the convergence of second order derivatives. Choosing n large enough yields the required contradiction. In what follows, we will hence look for a function h ∈ L 2 (Ω) satisfying (2.17) and´Ω h = 0.
According to (2.16), by using (2.15) and (2.12), there exist two positive constants A 1 and A 2 such thatF To obtain a contradiction, it hence suffices to construct a perturbation h ∈ L 2 (Ω) with support iñ Ω satisfying´Ω h = 0 and such thatˆΩ Let us prove that such a perturbation h exists. To this aim, let us introduce the operator L defined by This operator is self-adjoint and of compact inverse in L 2 (Ω), as a consequence of the spectral estimate (2.9). As a consequence, there exists a sequence of eigenvalues each of these eigenvalues being associated with a L 2 -normalised eigenfunction ψ k solving Let us fix K ∈ IN\{0} that will be fixed later and consider the family of linear functionals for every k ∈ 1, K . Let us define E k := ker(R k ) for every k ∈ 0, K . Observe that each space E k is of codimension at most 1. In particular, is of codimension at most (K + 1) in L 2 (Ω) and is non-empty. Let us hence pick F K ∈ E\{0} and assume by homogeneity, thatˆΩ Let us extend F K to Ω by setting F K = F K 1Ω. According to the definition of F K it follows that (ii) ∀k ∈ 0, K , one has´Ω(F k )1Ωθ m,µ = 0 so that, defining h K := F K 1Ω we have that h K is supported inΩ, belongs to L 2 (Ω) and satisfies´Ω h K = 0. Furthermore, the function η K = −F K 1Ωθ m,µ expands as Finally, we observe that, for this perturbation h K ,θ m,µ solves Using the L 2 (Ω)-orthogonality property of the eigenfunctions, we get and, similarly, We infer the existence of M > 0 such that The conclusion follows by taking K large enough so that λ K+1 (L m ) > M , which concludes the proof.

Comparison with the results of [25]
This section is dedicated to an explanation of the main difference with the proof of [25]. As recalled in Remark 4, the main result of [25] reads: ifΩ = {0 < m < 1} has an interior point, then it cannot be a solution of Problem (P µ ).
Although they do not use the expression (2.16) but an alternative expression of the second order Gâteaux-derivativeF µ , their idea, to reach a contradiction, is to reason backwards, by finding a function ψ, that "should" act asθ m,µ , well chosen to yield a contradiction, and then constructing an admissible perturbation h supported in the interior of {0 < m < 1} such thatθ m,µ = ψ.
However, we think that the authors, when carrying out their reasoning to reach a contradiction, made a slight mistake in the last line of the second set of equalities on [25,Page 11]. Indeed, it seems to us that it is implicitly assumed that ifθ m,µ is compactly supported inΩ, then so isθ m,µ , which is in general wrong.
Nevertheless, we propose hereafter an alternative proof of their result that uses their idea of first fixing a desirable function ϕ, and then proving the existence of an admissible perturbation h, compactly supported inΩ = {0 < m < 1} such that ϕ =θ m,µ , leading to a positive second order derivative.
Let us argue by contradiction, considering a solution m of (P µ ) such that the set has a non-empty interior (in particular, it is of positive measure). As a consequence of (2.4) there exists c such that θ m,µ p m,µ = c inΩ. Let χ ∈ D(IR d ) be a C ∞ , radially symmetric, non-negative function with compact support in B(0; r) such that χ(0) = 1. For every k ∈ IN, let us introduce ψ k defined by Since, by construction ψ k ∈ W 2,∞ (Ω) and since inf Ω θ m,µ > 0 we get that h k ∈ L ∞ (Ω). Moreover, since χ is compactly supported inΩ, so is h k . SinceΩ = {0 < m < 1}, the only condition we have to check to ensure that h k is admissible at m is that where the last equality comes from (2.31). Since by construction,´Ω ψ k = 0 the conclusion follows and hence h k is an admissible perturbation. Since sin 2 (k·) converges weakly to 1 2 in L 2 (0, r), since χ(0) = 1 and χ C 1 M for some M > 0, it follows that

Now, it remains to prove that
Finally, (I 3,k ) remains bounded and we get B(x0;r) |∇ψ k | 2 ∼ k→∞ k 2 C 0 for some constant C 0 > 0, which concludes the proof.

Remark 14.
In this approach which, as we underline, works under the strong hypothesis thatΩ has a non-empty interior, the core point is to build a sequence of admissible perturbations {h k } k∈IN such that the family H = {h k } k∈IN is uniformly bounded in W −2,2 but not in W −1,2 ; this guarantees the blow-up of the W 1,2 -norm and the boundedness of the L 2 -norm of the associated Gâteauxderivativesθ m,µ [h k ]. In the proof of Theorem I, the perturbation h that we construct has a fixed L 2 norm, and hence the sequence of Gâteaux-derivatives is uniformly bounded in W 2,2 (Ω).

Proof of Theorem II
The proof of Theorem II follows the same lines as the one of Theorem I. For this reason, we only indicate hereafter the main steps, and point to the principal differences.
Following the same methodology for stating the first order optimality conditions for problem (P µ ), let us introduce the adjoint state p j,m,µ solution of for every m ∈ M(Ω) and any admissible perturbation h at m. Let us compute the second order Gâteaux derivative of J j . Keeping track of the fact thatθ m,µ solves (2.5) and that by direct computation, we obtain we get an expression analogous to (2.16). Indeed, multiplying (2.43) byθ m,µ and integrating by parts yields Let us introduce u j,m,µ := p m,j,µ θ m,µ . (2.47) Notice that, using the same arguments as in the proof of Theorem I, we obtain inf Ω u j,m,µ > 0, ∆u j,m,µ ∈ L ∞ (Ω). (2.48) Since j belongs to C 2 , there exists M j > 0 such that We thus obtain the existence of a potential V j,m,µ ∈ L ∞ (Ω) such thaẗ We are now back to proving (2.19), and the proof reads the same way.

Proof of Theorem III
The core of this proof relies on fine energy estimates. In what follows, it will be convenient to introduce the set of bang-bang functions To alleviate the reading, let us start with the presentation of the proof structure.

Main idea
The proof rests upon the use of two ingredients: (i) the first one reads Lemma 15 ([22,Lemma 2]). There exists δ > 0 such that (ii) the second one, on which the emphasis will be put throughout the proof, is an estimate of the following form: there exist a constant M > 0 and two exponents α, β > 0 such that If Estimate (3.2) holds, then, assuming that the optimiser m * µ is a BV (Ω)-function (if it is not, then m BV (Ω) = +∞ and the statement of the theorem is trivial) then we have according to Lemma 15, yielding that To obtain convergence rates such as (3.2), we will proceed using energy arguments and prove that a rescaled, shifted version of the natural energy associated with the PDE (E m,µ ) yields this kind of control.
The rescaled energy functional Let us first recall that the equation (E m,µ ) admits a variational formulation: let us introduce then θ m,µ is characterized as the unique minimiser of E µ over W 1,2 (Ω); in other words Since we could not locate this formulation in the literature, we give a proof: over the set K := {u ∈ W 1,2 (Ω), u 0 in Ω}.
For the sake of completeness, this lemma is proved in Appendix A Let us introduceẼ Remark 17. The definition ofẼ m,µ is justified by the following, formal computation: let us assume that m is a C 1 function. It is known [16] that θ m,µ → µ→0 + m in L p (Ω), for p ∈ [1; +∞). Since we aim at obtaining a convergence rate for θ m,µ − m L 1 (Ω) as µ → 0 + , it is natural to consider the energy E m,µ (m). Explicit computations show that which justifies to consider the energyẼ m,µ .
Estimating θ m,µ − m L 1 (Ω) usingẼ m,µ . The key point is then to prove that θ m,µ − m L 1 (Ω) can be estimated in terms of the rescaled energy, via the following two Lemmas.

(3.8)
Proof of Lemma 18. We split the proof into two steps.
Step 1. There exists M > 0 such that This follows from explicit computations. Setting A =´Ω θm,µ Step 2. There exists M 0 > 0 such that for every µ > 0 and m ∈ M(Ω), one has According to the Jensen inequality, one has θ m,µ − m L 1 (Ω) |Ω| According to the Jensen inequality, there exists a constant C > 0 such that .
Since 0 |θ m,µ − m| 2 a.e. in Ω, this last inequality yields the existence of C ′ > 0 such that As a consequence, there exists C ′′ > 0 such that , whence the conclusion. The lemma follows from a combination of the two steps.
As a consequence, we will aim at proving an estimate of the form which, with Lemma 18, will lead to an estimate of the type (3.2). As explained in the introduction, we provide in Section B an alternative proof of (3.12) in the one-dimensional case following the method of Modica [23], that cannot unfortunately be straightforwardly extended to higher dimensions.
Let us now concentrate on the multidimensional case, assuming that Ω = (0; 1) d with d 1. We aim at obtaining an estimate of the form (3.2) which, by Lemma 18 amounts to determine a bound onẼ m,µ (θ m,µ ). Let us first consider m ∈ M(Ω) ∩ W 1,2 (Ω), that we will use as a test function in the energy. We getẼ We now consider a convolution kernel defined with the help of an approximation of unity. Namely, we consider a C ∞ function χ with compact support in B(0; 1) satisfying 0 χ 1 a.e. in B(0; 1),ˆB For every ε > 0, we define (3.14) Every m ∈ M(Ω) is extended outside of Ω by a compactly supported function of bounded variation, according to [1,Proposition 3.21]. We define for every ε > 0, where ⋆ stands for the convolution product in L 1 (IR d ). It is standard that According to [16,Equation (2.4)], there exists a constant M such that for every m, m ′ ∈ {f ∈ L ∞ (Ω), f 0}, there holds By the triangle inequality, for any m ∈ M(Ω) ∩ W 1,2 (Ω) and any µ, ε > 0, Note that one has, for any i ∈ 1, d , Hence, there exists M > 0 such that .

(3.19)
It follows that there exists M > 0 such that To end the proof, we need the following lemma, whose proof is postponed to the end of this section for the sake of clarity. We infer the existence of C 0 > 0 such that for all solution m * µ of (P µ ), one has and therefore, there exists c 0 > 0 such that, for every µ > 0 small enough The desired conclusion follows Proof of Lemma 19. As is customary in convolution, one has for a.e. x ∈ Ω, where τ h stands for the translation operator. However, we claim that It suffices to prove (3.24) for m ∈ C 1 , and the general result follows from the density of C 1 functions in BV (IR d ). For any h ∈ IR d we havê The desired result follows.

Conclusion: possible extensions of the bang-bang property to other state equations
We conclude this article with a discussion on possible generalisations of our method. Indeed, an interesting question is to know whether or not the methods put forth in the proof of Theorem I could be applied to other types of boundary conditions, for instance Dirichlet or Robin, or for other kinds of non-linearities. We justify below that it is the case, and that the main difficulty lies in the well-posedness of the equation acting as a constraint on the optimisation problem (P µ ). Let us consider a boundary operator B, that may be of Neumann (Bu = ∂u ∂ν ) or of Robin type (Bu = ∂u ∂ν + βu for some β > 0). Let us consider a non-linearity F = F (x, u) of class C 2 , and consider, for a given m ∈ M(Ω), the solution u m of The first assumption on F one has to make is: For any m ∈ M(Ω), (4.1) has a unique positive solution u m . Furthermore, inf It is notable that (H) is satsfied whenever F satisfies: 1. F (x, 0) = 0 and the steady state z(·) = 0 is unstable.  Given the assumption on j, (H ′′ ) is for instance implied if the first eigenvalue of −∆−m− ∂F ∂u (·, u m ) is positive (linear stability condition), that also ensures the Gâteaux differentiability of m → u m . This in turn holds if F (x, u) = −ug(x, u) with g∈ W 2,∞ non-decreasing. Using the adjoint state to computeJ(m)[h, h] in a more tractable form, one has Let us set Ψ m := pm um and V m : and the assumptions we made on F allow us to conclude that V ′ m belongs to L ∞ and that inf Ω Ψ m > 0. To obtain the bang-bang property, we argue by contradiction and assume that the setΩ := {0 < m * < 1} is of positive measure. To reach a contradiction, it suffices to exhibit a perturbation h that is supported inΩ such that Following the proof of Theorem I, we introduce the sequence of eigenfunctions and eigenvalues {ϕ k , λ k } k∈IN associated to the operator with Bϕ k = 0. Adapting hence the proof of Theorem I, we show that for any K ∈ IN, there exists an admissible perturbation h such that It follows that for such a perturbation, (4.10) infΩ Ψm yields the expected conclusion.

A Proof of Lemma 16
Let us first recall that since θ m,µ is non-negative and does not vanish in Ω, we have so that u(·) = 0 is not a minimiser of E m,µ . In order to prove this Lemma, let us introduce the energy functional In particular, if F m,µ has a minimiser u * , then |u * | also minimises F m,µ , and |u * | solves Conversely, if u * 0 is a minimiser of E m,µ then for any z ∈ W 1,2 (Ω), and so u * is a minimiser of F m,µ . Let us then prove that θ m,µ is a minimiser of F m,µ . Consider a minimising sequence {y k } k∈IN of F m,µ . Up to replacing y k with |y k | which, thanks to (A.3), would still yield a minimising sequence, we can assume that for every k ∈ IN, y k is non-negative. Let us introduce λ(m) as the first eigenvalue of the operator −∆ − m with Neumann boundary conditions. According to the Courant-Fischer principle, one has λ(m) = inf µˆΩ |∇u| 2 −ˆΩ mu 2 (A. 6) and therefore F m,µ (y k ) λ(m) y k 2 L 2 (Ω) + Since the embedding L 3 (Ω) ֒→ L 2 (Ω) is continuous, there exists C > 0 such that As a consequence, {y k } k∈IN is bounded in L 3 (Ω) and then also in L 2 by using the same argument. Finally, by definition of F m,µ , it is also uniformly bounded in W 1,2 (Ω). Hence, there exists a strong L 2 (Ω), weak L 3 (Ω) and weak W 1,2 closure point y ∞ ∈ W 1,2 (Ω) of {y k } k∈IN . Since the map IR ∋ x → |x| 3 is convex, the map L 3 (Ω) ∋ y →´Ω |y| 3 is lower semi-continuous. Hence it follows that lim inf k→∞ F m,µ (y k ) F m,µ (y ∞ ), (A.9) and y ∞ minimises F m,µ over K . Since 0 L 2 (Ω) is not a minimiser, we have y ∞ 0 and y ∞ (·) = 0.
The map x → |x| 3 is C 1 and the Euler-Lagrange equation on y ∞ writes    ∆y ∞ + y ∞ (m − y ∞ ) = 0 in Ω, ∂y∞ ∂ν = 0 in ∂Ω, y ∞ 0. (A.10) From uniqueness for non-zero, non-negative solutions of the logistic-diffusive PDE, it follows that y ∞ = θ m,µ . As a consequence: E m,µ (θ m,µ ) = F m,µ (θ m,µ ) = min B Proof of (3.12) in the one-dimensional case We assume in this section that Ω = (0, 1). Let us prove (3.12). The proof relies on ideas by Modica [23]. We can now prove Theorem 1. First of all, the maximiser m * µ of (P µ ) is a bang-bang function by Theorem I and belongs therefore to M(Ω). We thus obtain where δ > 0 is given by Lemma 15. The conclusion follows.
Proof of Proposition 20. In what follows, we will bypass the distinction between the interior perimeter of a subset A ⊂ (0; 1), denoted Per int (A), and its perimeter denoted Per(A) when seen as a subset of IR. Since we have obviously Furthermore, since we know from [22] that the BV (Ω) norm of maximisers blows-up as µ → 0, we can always assume that the set of finite perimeter A we are working with satisfies Per int (A) 2.
Since m ∈ M M (Ω), we know that m writes m = 1 A where A is a set of bounded perimeter. Let us then consider such a subset A. Since A is of finite perimeter, it writes with 0 a i < b i < a i+1 1 for every i ∈ 1, . . . , n .
To obtain the conclusion of the Proposition, it suffices to exhibit a constant C 1 that does not depend on µ, m, and a function u µ ∈ W 1,2 (Ω) such that E m,µ (u µ ) C 1 √ µ Per(A) .

(B.7)
Let us introduce h A , the so-called signed-distance function to the set A, defined by  for some regularization parameter ε 0, where η ε = ε 1 4 . We combine these two functions and introduce u ε = φ ε • h A . Let us use u ε as a test function in the variational formulation (3.5). We will estimate separately the gradient term and the remainder term of the energy functional.
Estimate of the gradient term. Since h A is differentiable a.e. and |h ′ A | = 1, we have (u ′ ε ) 2 = φ ′ ε (h A (·)) 2 a.e. in Ω. Using the decomposition of A, we get (B.10) Let us focus on the term The main interest of this decomposition is that on each interval (a i ; b i ) or (b i ; a i+1 ), the function h is symmetric with respect to the midpoint of the interval. As a consequence, two cases may occur when considering the interval (a i ; b i ) (the case (b i ; a i+1 ) being exactly identical): (i) either |b i − a i | 2η ε , in which case, since φ ′ ε L ∞ = 1 ηε it follows that As such, we have