Optimal location of resources maximizing the total population size in logistic models

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Journal de Mathématiques Pures et Appliquées

www.elsevier.com/locate/matpur

Optimal location of resources maximizing the total population size in logistic models

Idriss Mazari^a, Grégoire Nadin^b, Yannick Privat^c,∗

a SorbonneUniversités,UPMCUnivParis06,UMR7598,LaboratoireJacques-LouisLions,F-75005, Paris,France

bCNRS,SorbonneUniversités,UPMCUnivParis06,UMR7598,LaboratoireJacques-LouisLions, F-75005,Paris,France

cIRMA,UniversitédeStrasbourg,CNRSUMR7501,7rueRenéDescartes,67084Strasbourg,France

a r t i cl e i n f o a b s t r a c t

Articlehistory:

Received16May2018 Availableonlinexxxx

MSC:

49K20 35Q92 49J30 34B15

Keywords:

Diffusivelogisticequation Rearrangementinequalities Symmetrization

Optimalcontrol Shapeoptimization Optimalityconditions

Inthis article, weconsider a species whose populationdensity solvesthe steady diffusivelogisticequationinaheterogeneousenvironmentmodeledwiththehelpof aspatiallynonconstantcoefficientstandingforaresourcesdistribution.Weaddress the issue of maximizing the total population size with respect to the resources distribution,consideringsomeuniformpointwiseboundsaswellasprescribingthe totalamountofresources.Byassumingthediffusionrateofthespecieslargeenough, weprovethatanyoptimalconfigurationisbang-bang(inotherwordsanextreme point oftheadmissible set)meaning thatthisproblem can berecastas a shape optimizationproblem,theunknowndomainstandingfortheresourceslocation.In theone-dimensionalcase, thisproblem isdeeplyanalyzed,andforlargediffusion rates,all optimalconfigurationsareexhibited.Thisstudyiscompletedby several numericalsimulationsintheonedimensionalcase.

©2019ElsevierMassonSAS.Allrightsreserved.

r é s u m é

Dans cet article, nous considérons une espèce dont la densité de population résoutl’équationlogistiquediffusivestationairedansunenvironnementhétérogène modélisé à l’aide d’un coefficient non constant en espace représentant une distributionderessources.Nousabordonslaquestiondelamaximisationdelataille totaledelapopulationparrapportàlarépartitiondesressources,enconsidérant des contraintes ponctuelles uniformes et en prescrivant la quantité totale des ressources. En supposant que le taux de diffusion de l’espèce est suffisamment grand, nous prouvons que toute configuration optimale est bang-bang (c’est-à- dire unpointextrêmaldel’ensemble admissible),ce quisignifie que ceproblème peut être transformé en problème d’optimisation de forme, le domaine inconnu représentantl’emplacementdesressources.Danslecasunidimensionnel,ceproblème estanalyséprécisément,ainsiquepourdesvitessesdediffusionélevées,toutesles

✩ TheauthorswerepartiallysupportedbytheProject“Analysisandsimulationofoptimalshapes–applicationtolifesciences”

oftheParisCityHall.

✩✩ I. Mazari andY. Privat were partially supportedby the ANR Project SHAPO (ANR-18-CE40-0013– SHAPO – SHAPE OPTIMIZATION(2018)).

* Correspondingauthor.

E-mailaddresses:idriss.mazari@sorbonne-universite.fr(I. Mazari),gregoire.nadin@sorbonne-universite.fr(G. Nadin), yannick.privat@unistra.fr(Y. Privat).

https://doi.org/10.1016/j.matpur.2019.10.008

0021-7824/©2019ElsevierMassonSAS.Allrightsreserved.

configurations optimales sont exhibées. Cette étude est complétée par plusieurs simulationsnumériquesdanslecasunidimensionnel.

©2019ElsevierMassonSAS.Allrightsreserved.

1. Introduction

1.1. Motivationsandstateof theart

Inthisarticle,weinvestigateanoptimalcontrolproblemarisinginpopulationdynamics.Letusconsider the populationdensityθ_m,μ ofagivenspeciesevolvinginabounded andconnected domainΩ in Rⁿ with n ∈ N^∗, having a_C² boundary.In whatfollows, wewill assumethat θ_m,μ is thepositive solutionof the steady logistic-diffusiveequation (denoted(LDE) inthesequel) whichwrites

μΔθm,μ(x) + (m(x)−θm,μ(x))θm,μ(x) = 0 x∈Ω,

∂θm,μ

∂ν = 0 x∈∂Ω, (LDE)

where m ∈L^∞(Ω) stands for the resources distribution and μ >0 stands for thedispersal abilityof the species,alsocalleddiffusionrate.Fromabiologicalpointofview,therealnumberm(x) isthelocalintrinsic growthrateofspeciesatlocationxofthehabitatΩ andcanbeseenasameasureoftheresourcesavailable at x.

As will be explained below,we will only consider non-negative resourcedistributions m, i.e. such that m ∈L^∞₊(Ω) ={m∈L^∞(Ω), m≥0 a.e}.Inviewof investigatingthe influenceofspatial heterogeneityon themodel,weconsider theoptimalcontrolproblemofmaximizing thefunctional

Fμ:L^∞₊(Ω)m→

Ω

θm,μ,

wherethenotationffl

denotestheaverageoperator,inotherwordsffl

Ωf = _|Ω¹_|´

Ωf.ThefunctionalF stands for thetotal populationsize,inorder to furtherourunderstandingof spatial heterogeneityonpopulation dynamics.

In theframeworkofpopulation dynamics,thedensityθ_m,μ solvingEquation(LDE) canbe interpreted as asteadystateassociatedto thefollowingevolutionequation

⎧⎪

⎨

⎪⎩

∂u

∂t(t, x) =μΔu(t, x) +u(t, x)(m(x)−u(t, x)) t >0, x∈Ω

∂u

∂ν(t, x) = 0 t >0, x∈∂Ω

u(0, x) =u⁰(x)≥0, u⁰= 0 x∈Ω

(LDEE)

modelingthespatiotemporalbehaviorofapopulationdensityuinadomainΩ withthespatiallyheteroge- neousresourcetermm.

ThepioneeringworksbyFisher[1],Kolmogorov-Petrovski-Piskounov [2] and Skellam[3] onthelogistic diffusive equation weremainly concerned withthe spatiallyhomogeneous case.Thereafter, many authors investigated the influence of spatial heterogeneity on population dynamics and species survival. In [4], propagation properties ina patch model environment are studied. In [5], aspectral condition for species survivalinheterogeneousenvironmentshasbeenderived,while[6] dealswiththeinfluenceoffragmentation and concentrationof resources onpopulation dynamics.These workswerefollowedby [7] dedicatedto an optimaldesignproblem,thatwill becommentedinthesequel.

Investigating existence anduniquenessproperties ofsolutionsfor thetwo previousequations as well as theirregularity propertiesboils downtothestudyofspectral propertiesforthelinearizedoperator

L:D(L)f →μΔf+mf,

wherethedomainofLisD(L)={f ∈L²(Ω)|Δf ∈L²(Ω)}andofitsfirsteigenvalueλ1(m,μ),character- izedbytheCourant-Fischerformula

λ1(m, μ) := sup

f∈W^1,2(Ω),´

Ωf²=1

⎧⎨

⎩−μ ˆ

Ω

|∇f|²+ ˆ

Ω

mf²

⎫⎬

⎭. (1) Indeed,thepositivenessofλ(m,μ) isasufficientconditionensuringthewell-posednessofequations(LDEE) and (LDE) ([7]). Then, Equation(LDE) has auniquepositivesolution θm,μ ∈W^1,2(Ω).Furthermore,for anyp≥1,θ_m,μ belongstoW^2,p(Ω),and thereholds

0<inf

Ω

θm,μ≤θm,μ≤ mL^∞(Ω). (2)

Moreover, thesteady stateθm,μ is globallyasymptotically stable:for anyu0 ∈W^1,2(Ω) such thatu0 ≥0 a.e.inΩ andu0= 0,onehas

u(t,·)−θ_m,μL^∞(Ω)_t→+∞−→ 0,

where udenotes the unique solutionof (LDEE) with initial state u0 (belonging to L²(0,T;W^1,2(Ω)) for everyT >0).

The importance of λ₁(m,μ) forstability issues related to population dynamics models was first noted in simple cases by Ludwig, Aronson and Weinberger [8]. Let us mention [9] where the case of diffusive Lotka-Volterraequationsisinvestigated.

To guarantee that λ₁(m,μ)> 0, it is enough to work with distributions of resources m satisfying the assumption

m∈L^∞₊(Ω) where L^∞₊(Ω) =

⎧⎨

⎩m∈L^∞(Ω), ˆ

Ω

m >0

⎫⎬

⎭. (H1) Notethattheissueofmaximizingthisprincipaleigenvaluewasaddressedforinstancein[10–14].

Inthe survey article[15], Lou suggeststhefollowing problem: theparameter μ>0 beingfixed, which weightmmaximizesthetotalpopulation sizeamong alluniformly boundedelementsof L^∞(Ω)?

Inthis article, weaim at providingpartial answersto this issue,and moregenerallynew resultsabout theinfluenceof thespatial heterogeneitym(·) onthetotalpopulationsize.

Forthatpurpose,letusintroducethetotalpopulation sizefunctional,definedforagivenμ>0 by Fμ:L^∞₊(Ω)m−→

Ω

θm,μ, (3)

whereθ_m,μ denotesthesolutionofequation(LDE).

Letus mentionseveral previousworksdealingwiththemaximizationof thetotalpopulation size func- tional.Itisshownin[16] that,amongallweightsmsuchthatffl

Ωm=m0,thereholds Fμ(m)≥ Fμ(m0)= m₀; this inequality is strict whenever m is nonconstant. Moreover, it is also shown that the problem of maximizingFμ overL^∞₊(Ω) has nosolution.

Remark 1.The fact that m ≡ m₀ is a minimum for Fμ among the resources distributions m satisfying ffl

Ωm=m0relies onthefollowingobservation:multiplying(LDE) by _θ¹

m,μ and integratingbypartsyields μ

Ω

|∇θm,μ|² θm,μ2 +

Ω

(m−θm,μ) = 0 (4)

and therefore, Fμ(m)=m₀+μ´

Ω

|∇θm,μ|²

θm,μ2 ≥m₀ =Fμ(m₀) forall m∈ L^∞₊(Ω) such thatffl

Ωm =m₀. It follows thattheconstantfunctionequaltom₀isaglobalminimizerofFμ over{m∈L^∞₊(Ω),ffl

Ωm=m₀}. Intherecentarticle[17],itisshownthat,whenΩ= (0,),onehas

∀μ >0, ∀m∈L^∞₊(Ω)|m≥0 a.e.,

Ω

θ_m,μ ≤3

Ω

m.

This inequalityissharp, althoughtheright-handsideisneverreached,andtheauthorsexhibit asequence (mk,μk)_k∈N suchthatffl

Ωθmk,μk/ffl

Ωmk→3 ask→+∞,butforsuchasequencethereholdsmkL^∞(Ω)→ +∞andμ_k →0 ask→+∞.

In[15],itisprovedthat,withoutL¹ orL^∞boundsontheweightfunctionm,themaximizationproblem is ill-posed. It isthus naturalto introducetwo parametersκ,m0 >0,and to restrict ourinvestigation to theclass

Mm0,κ(Ω) :=

⎧⎨

⎩m∈L^∞(Ω),0≤m≤κa.e ,

Ω

m=m₀

⎫⎬

⎭. (5) It isnotablethatin[18], themoregeneralfunctionalJ_B definedby

J_B(m) = ˆ

Ω

(θ_m,μ−Bm²) forB ≥0

is introduced.Inthe caseB= 0, theauthorsapply theso-calledPontryagin principle,showtheGâteaux- differentiabilityofJ_B andcarryoutnumericalsimulationsbackinguptheconjecturethatmaximizersofJ₀ over Mm0,κ(Ω) areofbang-bangtype.

However, proving this bang-bang property is a challenge. The analysis of optimal conditions is quite complex, because the sensitivity of the functional “total population size” with respect to m(·) isdirectly related to thesolution of anadjoint state, solvinga linearizedversion of (LDE).Deriving and exploiting the propertiesofoptimal configurationstherefore requires athoroughunderstandingof theθm,μ behavior as wellastheassociatedstate.Todothis,weareintroducinganewasymptoticmethodto exploitoptimal conditions.

Wewill investigatetwo propertiesofthemaximizersofthetotalpopulationsize functionFμ.

1. Pointwiseconstraints.Themainissuethatwillbeaddressedinwhatfollowsisthebang-bangcharacter ofoptimalweightsm^∗(·),inotherwords,whetherm^∗ isequalto0orκalmosteverywhere.Notingthat Mm0,κ(Ω) isaconvexset andthatbang-bang functionsare theextreme pointsofthisconvexset,this questionrewrites:

Arethemaximizersm^∗ extremepointsof thesetMm0,κ(Ω)?

Fig. 1.Ω = (0,1)². The left distribution is “concentrated” (connected) whereas the right one is fragmented (disconnected).

Inourmainresult(Theorem1)weprovideapositiveanswerforlargediffusivities.Itisnotablethatour proof restsuponawell-adaptedexpansionofthesolutionθm,μ of(LDE) withrespectto thediffusivity μ.

This approachcould be consideredunusual, since such resultsare usually obtainedby an analysis of the propertiesof theadjointstate (orswitching function). However,sincethe switchingfunction very implicitly dependsonthedesignvariablem(·),wedidnotobtainthisresultinthis way.

2. Concentration-fragmentation. Itis wellknownthatresourceconcentration(whichmeansthatthedis- tribution of resourcesm decreasesinalldirections,see Definition2foraspecific statement)promotes the survival of species[7]. On the contrary, we will saythat aresource distributionm =κχE, where E isasubsetofΩ,isfragmented whentheE set isdisconnected.IntheFig.1, Ω isasquare,and the intuitivenotionofconcentration-fragmentationofresourcedistributionisillustrated.

A naturalissuerelatedto qualitativepropertiesofmaximizersisthus Aremaximizersm^∗ concentrated? Fragmentated?

InTheorem2,weconsiderthecaseofaorthotopeshapehabitat,andweshowthatconcentrationoccurs forlargediffusivities:ifmμmaximizesFμoverMm0,κ(Ω),thesequence{mμ}μ>0stronglyconvergesin L¹(Ω) toaconcentrateddistributionas μ→ ∞.

In the one-dimensional case, we also prove thatif the diffusivity is large enough, there are only two maximizers, thatareplottedin Fig.2(seeTheorem 3).

Finally, inthe one-dimensionalcase, weobtain asurprising result:fragmentation may bebetter than concentrationforsmalldiffusivities(seeTheorem 4andFig.3).

This issurprising becauseinmanyproblemsofoptimizingthelogistic-diffusiveequation,itisexpected thatthebestdispositionofresourceswillbe concentrated.

1.2. Main results

Inthewholearticle, thenotation χ_I will be used todenote the characteristicfunction ofameasurable subsetI ofRⁿ,inotherwords, thefunctionequalto1inI and0elsewhere.

For the reasons mentioned inSection 1.1, it is biologically relevant to consider theclass of admissible weights Mm0,κ(Ω) defined by (5), where κ > 0 and m₀ > 0 denote two positive parameters such that m₀< κ (sothatthis setisnontrivial).

Wewillhenceforthconsiderthefollowing optimaldesignproblem.

Fig. 2.Ω = (0,1). Plot of the only two maximizers ofFμoverMm0,κ(Ω).

Fig. 3.Ω = (0,1). A double crenel (on the left) is better than a single one (on the right).

Optimaldesign problem. Fix n ∈N^∗, μ >0, κ>0, m₀ ∈ (0,κ) and let Ω be a bounded connected domain ofRⁿ having a_C² boundary. Weconsidertheoptimization problem

sup

m∈Mκ,m0(Ω)Fμ(m). (Pμⁿ)

As will be highlighted in thesequel, the existence of amaximizer follows from adirect argument. We will thusinvestigatethequalitativepropertiesofmaximizersdescribed intheprevioussection(bang-bang character,concentration/fragmentationphenomena).

Forthesakeofreadability,almostalltheproofsarepostponedtoSection2.

Let us stress that the bang-bang character of maximizer is of practical interest in view of spreading resources inanoptimalway.Indeed,inthecasewhereamaximizerm^∗ writesm^∗=κχ_E,thetotalsize of population ismaximizedbylocatingalltheresourcesonE.

1.2.1. Firstproperty

Westart withapreliminaryresultrelated tothe saturationofpointwise constraintsforProblem (Pμⁿ), valid for all diffusivities μ. It is obtained by exploitingthe first order optimality conditions for Problem (Pμⁿ),writtenintermsofanadjointstate.

Proposition1.Letn∈N^∗,μ>0,κ>0,m0∈(0,κ).Letm^∗ beasolutionof Problem (Pμⁿ).Then, theset {m=κ}∪ {m= 0}hasapositive measure

1.2.2. Thebang-bangproperty holdsforlarge diffusivities

For large values of μ, we will prove that the variational problem can be recast in terms of a shape optimizationproblem,as underlinedinthenextresults.

Theorem 1.Letn ∈N^∗,κ>0,m0 ∈ (0,κ). Thereexists a positivenumber μ^∗ =μ^∗(Ω,κ,m0) such that, forevery μ≥μ^∗,thefunctional Fμ isstrictly convex.Asaconsequence, forμ≥μ^∗,any maximizerof Fμ

overMm0,κ(Ω) (orsimilarlyany solution ofProblem (Pμⁿ))ismoreover of bang-bangtype.¹

WeemphasizethattheproofofTheorem1is quiteoriginal, sinceitdoesnotrestupontheexploitation ofadjointstateproperties,butupontheuseofapowerseriesexpansionsinthediffusivityμofthesolution θ_m,μof(LDE),aswellastheirderivativewithrespecttothedesignvariablem.Inparticular,thisexpansion isusedtoprovethat,ifμislargeenough,thenthefunctionFμ isstrictlyconvex.Sincetheextremepoints ofMm0,κ(Ω) arebang-bangresourcesfunctions,theconclusionreadilyfollows.

Thistheorem justifiesthatProblem(Pμⁿ) canbe recast asashapeoptimizationproblem.Indeed,every maximizerm^∗ isoftheform m^∗=κχE whereE isameasurablesubsetsuchthat|E|=m0|Ω|/κ.

Remark2.Wecanrewritethis resultintermsofshapeoptimization,byconsideringasmain unknownthe subset E of Ω where resources are located: indeed, under the assumptions of Theorem 1, there exists a positivenumberμ^∗=μ^∗(Ω,κ,m0) suchthat,foreveryμ≥μ^∗, theshape optimizationproblem

sup

E⊂Ω,|E|=m0|Ω|/κFμ(κχ_E), (6) where thesupremum istaken overallmeasurable subset E ⊂Ω such that|E|=m0|Ω|/κ,has asolution.

We underline the fact that, for such a shape functional, which is “non-energetic” in the sense that the solutionof thePDEinvolved cannotbe seenas aminimizerof thesamefunctional,proving theexistence ofmaximizersisusuallyintricate.

1.2.3. Concentrationoccurs forlargediffusivities

Inthissection,westatetworesultssuggestingconcentrationpropertiesforthesolutionsofProblem(Pμⁿ) mayholdforlargediffusivities.

Forthatpurpose,letusintroducethefunctionspace

X :=W^1,2(Ω)∩

⎧⎨

⎩u∈W^1,2(Ω),

Ω

u= 0

⎫⎬

⎭ (7)

andtheenergyfunctional

1 Inotherwords,itis,anelementofMm0,κ(Ω) equala.e.to0orκinΩ.

Em:X u→ 1 2

Ω

|∇u|²−m0 Ω

mu. (8)

Theorem 2.[Γ-convergenceproperty]

1. LetΩbe adomain witha _C² boundary.Forany μ>0,letm_μ bea solutionof Problem (Pμⁿ).Any L¹ closure pointof {m_μ}μ>0 asμ→ ∞ isasolution oftheoptimaldesignproblem

m∈Mminm0,κ(Ω)min

u∈XEm(u). (9)

2. In the case of atwo dimensional orthotope Ω= (0;a1)×(0;a2), any solution of theasymptotic opti- mization problem(9)decreases in everydirection.

As willbeclearintheproof,thistheorem isaΓ-convergence property.

Intheone-dimensionalcase,onecanrefinethisresultbyshowingthat,forμlargeenough,themaximizer is astepfunction.

Theorem 3.Letusassume thatn= 1 andΩ= (0,1).Letκ>0,m₀∈(0,κ).Thereexistsμ >ˆ 0suchthat, forany μ≥μ,ˆ anysolution m ofProblem (Pμⁿ)isequala.e.toeither m˜ orm(1˜ − ·),where m˜ =κχ(1−,1)

and =m0/κ.

1.2.4. Fragmentationmay occurforsmall diffusivities

Let us concludeby underlining thatthe statementof Theorem 3cannot be truefor allμ >0.Indeed, weprovideanexampleinSection3.1whereadouble-crenelgrowthrategivesalargertotalpopulationsize than thesimplecrenelm˜ ofTheorem3.

Theorem 4.The function m˜ =κχ_(1−,1) (and m(1˜ − ·)=κχ_(0,)) does not solve Problem (Pμⁿ) for small valuesof μ.Moreprecisely,if weextendm˜ outside of(0,1) by periodicity,there existsμ>0suchthat

Fμ

m(2˜ ·)

>Fμ( ˜m).

This result is quite unusual.For the optimization of the first eigenvalue λ1(m,μ) defined by (1) with respect tomontheinterval(0;1)

sup

m∈M((0;1))λ₁(m, μ)

we know(see[7])thattheonlysolutionsarem˜ andm(1˜ − ·),foranyμ.

It isnotablethatthefollowing resultisabyproductofTheorem 4above.

Corollary 1.Thereexistsμ>0suchthat theproblems sup

m∈M((0;1))λ₁(m, μ) and

sup

m∈M((0;1))Fμ(m) do not havethesamemaximizers.

Forfurthercommentsontherelationshipbetweenthemaineigenvalueandthetotalsizeofthepopulation, wereferto[19],whereanasymptoticanalysisofthemaineigenvalue(relativetoμasμ→+∞)isperformed, andthereferencestherein.

Weconcludethissectionbymentioningtherecentwork[20],thatwasreportedtouswhenwewrotethis article.Theyshowthat,ifweassumetheoptimaldistributionofregularresources(moreprecisely,Riemann integrable),thenitisnecessarilyofbang-bangtype.Theirproofisbasedonaperturbationargumentvalid forallμ>0.However,provingsuchregularityisgenerallyquitedifficult.Ourproof,althoughitisnotvalid forallμ,isnotbasedonsucharegularityassumption,butthesetwocombinedresultsseemtosuggestthat allmaximizersofthisproblemare ofbang-bangtype.

1.3. Tools andnotations

Inthissection,wegathersomeusefultoolswewilluseto provethemainresults.

Rearrangementsoffunctionsandprincipaleigenvalue Letusfirstrecallseveralmonotonicityandregularity relatedtotheprincipaleigenvalueoftheoperatorL.

Proposition2.[9] Let m∈L^∞₊(Ω) andμ>0.

(i) Themapping R^∗₊μ→λ₁(m,μ)iscontinuous andnon-increasing.

(ii) Ifm≤m₁,then λ₁(m,μ)≤λ₁(m₁,μ),andtheequality istrueif,and onlyif m=m₁ a.e.in Ω.

In the proof of Theorem 3, we will use rearrangement inequalities at length. Let us briefly recall the notionofdecreasingrearrangement.

Definition1.Foragivenfunctionb∈L²(0,1),onedefinesitsmonotonedecreasing(resp.monotoneincreas- ing)rearrangementbdr (resp.bbr)on(0,1) bybdr(x)= sup{c∈R|x∈Ω^∗_c},whereΩ^∗_c = (1− |Ωc|,1) with Ωc={b> c}(resp.bbr(·)=bdr(1− ·)).

The functions b_dr and b_br enjoy nice properties. In particular, the Pólya-Szegö and Hardy-Littlewood inequalitiesallowtocompareintegralquantitiesdependingonb,b_dr,b_br andtheirderivative.

Theorem([21,22]). Letubeanon-negative andmeasurable function.

(i) Ifψ isany measurablefunctionfrom R+ toR,then ˆ1

0

ψ(u) = ˆ1

0

ψ(udr) = ˆ1

0

ψ(ubr) (equimeasurability);

(ii) Ifubelongs toW^1,p(0,1)with 1≤p,then ˆ1

0

(u)^p≥ ˆ1

0

(u_br)^p= ˆ1

0

(u_dr)^p (Pólya inequality);

(iii) If u,v belong toL²(0,1),then ˆ1

0

uv≤ ˆ1

0

ubrvbr= ˆ1

0

udrvdr (Hardy-Littlewood inequality);

The equality case inthe Pólya-Szegö inequality is the object of the Brothers-Ziemertheorem (see e.g.

[23]).

Symmetric decreasingfunctions Inhigherdimensions,inorder tohighlightconcentrationphenomena,we will useanother notion of symmetry, namely monotone symmetric rearrangements thatare extensions of monotone rearrangementsinonedimension. Here,Ω denotesthen-dimensionalorthotopen

i=1(ai,bi).

Definition 2.Foragiven functionb∈L¹(Ω),onedefines itssymmetricdecreasingrearrangementb_sd onΩ as follows: first fix then−1 variablesx₂,. . . ,x_n. Define b_1,sd as the monotone decreasingrearrangement of x→b(x,x₂,. . . ,x_n).Then fixx₁,x₃,. . . ,x_n anddefine b_2,sd as themonotonedecreasingrearrangement of x→ b1,sd(x1,x,. . . ,xn). Performsuch monotone decreasingrearrangements successively. Theresulting functionisthesymmetricdecreasingrearrangementofb.

We define thesymmetric increasing rearrangementin adirection i asimilar fashion and write it bi,id. Note that,inhigherdimensions,thedefinitionofdecreasingrearrangementstrongly dependsontheorder inwhichthevariablesaretaken.

Similarly to the one-dimensional case, the Pólya-Szegö and Hardy-Littlewood inequalities allow us to compare integralquantities.

Theorem ([7,24]). Letubeanon-negative andmeasurablefunction definedon aboxΩ=n

i=1(0;ai).

(i) If ψ isanymeasurable functionfromR+ toR,then ˆ

Ω

ψ(u) = ˆ

Ω

ψ(usd) = ˆ

Ω

ψ(usd) (equimeasurability);

(ii) If u belongs to W^1,p(Ω) with 1 ≤ p, then, for every i ∈ NN, [ai;bi] = ω1,i∪ω2,i∪ω3,i, where the map (x1,. . . ,xi−1,xi+1,. . . ,xN)→u(x1,. . . ,xn)is decreasing if xi ∈ω1,i,increasing if xi ∈ω2,i and constant ifxi∈ω3,i.

ˆ

Ω

|∇u|^p≥ ˆ

Ω

|∇u_sd|^p (Pólya inequality);

Furthermore, if, for any i ∈ Nn, ´

Ω|∇u|^p = ´

Ω|∇ui,sd|^p then there exist three measurable subsets ω_i,1,ω_i,2 andω_i,3 of(0;a_i)such that

1. (0;a_i)=ω_i,1∪ω_i,2ω_i,3, 2. u=u_i,bd oni−1

k=1(0;a_k)×ω_i,1×n

k=i+1(0;a_k), 3. u=u_i,id on_i−1

k=1(0;a_k)×ω_i,2×n

k=i+1(0;a_k), 4. u=ui,bd=ui,id on _i−1

k=1(0;ak)×ωi,3×n

k=i+1(0;ak).

(iii) If u,v belongtoL²(Ω),then ˆ

Ω

uv≤ˆ

Ω

usdvsd (Hardy-Littlewood inequality);

Poincaréconstantsandellipticregularityresults Wewilldenotebyc^(p) theoptimalpositiveconstantsuch thatforeveryp∈[1,+∞),f ∈L^p(Ω) andu∈W^1,p(Ω) satisfying

Δu=f inD(Ω),

thereholds

uW^2,p(Ω)≤c^(p)

fL^p(Ω)+uL^p(Ω)

.

The optimal constant in the Poincaré-Wirtinger inequality will be denoted by C_{P W}^(p) (Ω). This inequality reads: foreveryu∈W^1,p(Ω),

u−

Ω

u L^p(Ω)

≤C_{P W}^(p) (Ω)∇uL^p(Ω). (10)

Wewillalsousethefollowingregularity results:

Theorem. ([25, Theorem 9.1]) Let Ω be a C² domain. There exists a constant C_Ω > 0such that, if f ∈ L^∞(Ω) andu∈W^1,2(Ω) solve

−Δu=f in Ω,

∂u

∂ν = 0 on ∂Ω, (11)

then

∇uL¹(Ω)≤C_ΩfL¹(Ω). (12) Theorem. ([26, Theorem 1.1]) Let Ω be a C² domain. There exists a constant CΩ > 0such that, if f ∈ L^∞(Ω) andu∈W^1,2(Ω) solve

−Δu=f in Ω,

∂u

∂ν = 0 on ∂Ω, (13)

then

∇uL^∞(Ω)≤C_ΩfL^∞(Ω). (14) Remark3.Thisresultisinfactacorollary[26,Theorem1.1].InthisarticleitisprovedthattheL^∞norm ofthegradientoff isboundedbytheLorentznormL^n,1(Ω) off,whichisautomaticallycontrolledbythe L^∞(Ω) normoff.

Note that Stampacchia’s original result deals with Dirichlet boundary conditions. However, the same dualityargumentsprovidetheresultforNeumannboundaryconditions.

Theorem. ([27]) Let r ∈ (1;+∞). There exists Cr > 0 such that, if f ∈L^r(Ω) and if u∈ W^1,r(Ω) be a solution of

−Δu= div(f) inΩ,

∂u

∂ν = 0 on∂Ω, (15)

thenthere holds

∇uL^r(Ω)≤C_rfL^r(Ω). (16)

2. Proofsofthe mainresults

2.1. Firstorderoptimalityconditions forProblem (Pμⁿ)

To provethemain results,wefirstneedtostatethefirstorder optimalityconditionsforProblem(Pμⁿ).

Forthatpurpose,letusintroducethetangentconetoMm0,κ(Ω) atanypointofthis set.

Definition 3.([28, chapter 7]) For every m ∈ Mm0,κ(Ω), the tangent cone to the set Mm0,κ(Ω) at m, denoted by Tm,Mm0,κ(Ω) is the set of functions h ∈ L^∞(Ω) such that, for any sequence of positive real numbersε_n decreasingto 0,there existsasequenceoffunctionsh_n∈L^∞(Ω) converging tohasn→+∞, and m+ε_nh_n∈ Mm0,κ(Ω) foreveryn∈N.

We will show that, for any m ∈ Mm0,κ(Ω) and any admissible perturbation h ∈ Tm,Mm0,κ(Ω), the functionalFμ istwiceGâteaux-differentiableatmindirectionh.Todothat,wewillshowthatthesolution mapping

S:m∈ Mm0,κ(Ω)→θ_m,μ∈L²(Ω),

where θ_m,μ denotes thesolutionof(LDE),istwiceGâteaux-differentiable.Inthisview,weprovide several L²(Ω) estimates ofthesolutionθm,μ.

Lemma 1.([18])The mapppingS is twiceGâteaux-differentiable.

For the sakeofsimplicity, we will denote by θ˙m,μ =dS(m)[h] theGâteaux-differentialof θm,μ at min directionhandbyθ¨m,μ=d²S(m)[h,h] itssecondorder derivativeat mindirectionh.

Elementary computationsshowthatθ˙m,μ solvesthePDE

μΔ ˙θm,μ+ (m−2θm,μ) ˙θm,μ =−hθm,μ in Ω,

∂θ˙m,μ

∂ν = 0 on∂Ω, (17)

whereasθ¨m,μ solvesthePDE

μΔ¨θm,μ+ ¨θm,μ(m−2θm,μ) =−2

hθ˙m,μ−θ˙²_m,μ

in Ω,

∂θ¨m,μ

∂ν = 0 on∂Ω. (18)

It followsthat,forallμ>0,theapplicationFμ isGâteaux-differentiablewithrespectto mindirection handitsGâteauxderivativewrites

dFμ(m)[h] = ˆ

Ω

θ˙m,μ.

SincetheexpressionofdFμ(m)[h] aboveisnotworkable,weneedtointroducetheadjointstatepm,μtothe equation satisfiedbyθ˙m,μ,i.ethesolutionoftheequation

μΔpm,μ+pm,μ(m−2θm,μ) = 1 in Ω,

∂pm,μ

∂ν = 0 on∂Ω. (19)

Note that p_m,μ belongs to W^1,2(Ω) and is unique, accordingto theFredholm alternative. Infact, we can prove thefollowing regularity resultson p_m,μ: p_m,μ ∈ L^∞(Ω),p_m,μL^∞(Ω) ≤M,where M isuniform in

m ∈ Mm0,κ(Ω) and, for any p ∈ [1;+∞), p_m,μ ∈ W^2,p(Ω), so that Sobolev embeddings guarantee that p_m,μ∈C^1,α(Ω).

Now, multiplying the main equation of (19) by θ˙m,μ and integrating two times by parts leads to the expression

dFμ(m)[h] =− ˆ

Ω

hθ_m,μp_m,μ.

Nowconsideramaximizerm.ForeveryperturbationhintheconeTm,Mm0,κ(Ω),thereholdsdFμ(m)[h]≥0.

Theanalysis ofsuch optimalitycondition isstandardinoptimalcontroltheory(see forexample[29]) and leadstothefollowingresult.

Proposition3.Letusdefine ϕm,μ =θm,μpm,μ,whereθm,μ andpm,μ solverespectivelyequations(LDE)and (19).Thereexistsc∈R suchthat

{ϕ_m,μ < c}={m=κ}, {ϕ_m,μ=c}={0< m < κ}, {ϕ_m,μ > c}={m= 0}. 2.2. Proof ofProposition 1

Aneasybuttediouscomputationshowsthatthefunctionϕ_m,μ introducedinProposition3is_C^1,α(Ω)∩ W^1,2(Ω) function,asaproductoftwo _C^1,α functionsandsatisfies(in aW^1,2 weaksense)

μΔϕm,μ−2μ

∇ϕm,μ,^∇_θ^θm,μ

m,μ

+ϕm,μ

2μ^|∇_θ^θm,μ|2

m,μ2 + 2m−3θm,μ

=θm,μ in Ω,

∂ϕm,μ

∂ν = 0 on∂Ω, (20)

where ·,· stands for the usual Euclidean inner product. To provethat |{m = 0}|+|{m = κ}|> 0,we argueby contradiction,byassuming that|{m=κ}|=|{m= 0}|= 0. Therefore,ϕm,μ =c a.e.inΩ and, accordingto(20),thereholds

c

2μ|∇θm,μ|²

θm,μ2 + 2m−3θ_m,μ

=θ_m,μ Integratingthisidentityand usingthatθm,μ >0 inΩ andc= 0, weget

2c

⎛

⎝μˆ

Ω

|∇θm,μ|² θm,μ2 +

ˆ

Ω

(m−θm,μ)

⎞

⎠= (c+ 1) ˆ

Ω

θm,μ.

Equation(4) yields thattheleft-handsideequals 0,sothatonehasc=−1.Comingbacktotheequation satisfiedbyϕ_m,μ leadsto

m=θ_m,μ−μ|∇θ_m,μ|² θ_m,μ² .

Thelogisticdiffusiveequation(LDE) isthentransformedinto μθm,μΔθm,μ−μ|∇θm,μ|²= 0.

Integrating this equation by parts yields ´

Ω|∇θ_m,μ|² = 0.Thus, θ_m,μ is constant,and so is m. In other words,m=m₀,which, accordingto(4) (seeRemark1)isimpossible.Theexpectedresultfollows.

2.3. Proofof Theorem1

TheproofofTheorem1isbasedonacarefulasymptoticanalysiswithrespecttothediffusivityvariable μ.

Letus firstexplaintheoutlinesoftheproof.

Let us fix m ∈ Mm0,κ(Ω) and h ∈ L^∞(Ω). In the sequel, the dot or double dot notation f˙ or f¨will respectivelydenote firstandsecondorder Gâteaux-differentialoff atm indirectionh.

According to Lemma 1, Fμ is twice Gâteaux-differentiable and its second order Gâteaux-derivative is given by

d²Fμ(m)[h, h] = ˆ

Ω

θ¨m,μ,

where θ¨m,μ isthesecondGâteauxderivativeofS,definedastheuniquesolutionof(18).

Letm1 andm2be twoelementsofMm0,κ(Ω) anddefine φ_μ: [0; 1]t→ Fμ

tm₂+ (1−t)m₁

−tFμ(m₂)−(1−t)Fμ(m₁).

Onehas

d²φ_μ dt² (t) =

ˆ

Ω

θ¨_{(1−t)m1+tm2,μ}, and φ_μ(0) =φ_μ(1) = 0,

where θ¨_{(1−t)m1+tm2,μ} mustbe interpretedas abilinear formfrom L^∞(Ω) toW^1,2(Ω),evaluatedtwo times at thesamedirectionm₂−m₁.Hence,togetthestrictconvexityofFμ,itsufficestoshowthat,whenever μ islargeenough,

ˆ

Ω

θ¨tm2+(1−t)m1,μ>0

as soon as m1 =m2 (in L^∞(Ω)) and t ∈ (0,1),or equivalently thatd²Fμ(m)[h,h] > 0 as soon as m ∈ Mm0,κ(Ω) andh∈L^∞(Ω).Notethatsinceh=m₂−m₁,itispossibletoassumewithoutlossofgenerality thathL^∞(Ω)≤2κ.

Theproofisbasedonanasymptoticexpansionofθm,μintoamaintermandareminderone,withrespect to thediffusivityμ.Itiswell-known(seee.g.[16,Lemma2.2])thatonehas

θ_m,μ ^W

1,2(Ω)

−−−−−−→

μ→∞ m₀. (21)

However,sinceweareworkingwithresourcesdistributionslivinginMm0,κ(Ω),suchaconvergenceproperty doesnotallowustoexploititforderivingoptimalitypropertiesforProblem(Pμⁿ).

Forthisreason,inwhatfollows,wefindafirstorderterminthisasymptoticexpansion.Togetaninsight into theproof’smain idea,letus firstproceedinaformalway, bylookingforafunctionη1,msuchthat

θm,μ ≈m0+η_1,m μ

as μ→ ∞.Pluggingthisformal expansionin(LDE) andidentifyingat order ¹_μ yieldsthatη_1,m satisfies

Δη_1,m+m₀(m−m₀) = 0 in Ω,

∂η1,m

∂ν = 0 on∂Ω.

To make this equation well-posed, it is convenient to introduce the function ˆη_1,m defined as the unique solutiontothesystem

⎧⎪

⎨

⎪⎩

Δˆη_1,m+m₀(m−m₀) = 0 in Ω,

∂ηˆ1,m

∂ν = 0 on∂Ω,

ffl

Ωηˆ_1,m= 0, andto determineaconstantβ_1,m suchthat

η1,m= ˆη1,m+β1,m.

Inviewofidentifyingtheconstantβ1,m,weintegrateequation(LDE) toget ˆ

Ω

θm,μ(m−θm,μ) = 0

whichyields,at theorder ¹_μ,

β1,m= 1 m₀

Ω

ˆ

η1,m(m−m0) = 1 m²₀

Ω

|∇ηˆ1,m|².

Therefore,onehasformally

Ω

θ_m,μ ≈m₀+1 μ

Ω

|∇ηˆ_1,m|².

As will be proved in Step 1 (paragraph 2.3), the mapping Mm0,κ(Ω) m → β1,m is convex so that, at the order _μ¹, the mapping Mm0,κ(Ω) m → ffl

Ωθm,μ is convex. We will prove the validity of all the claimsabove,bytakingintoaccountremaindertermsintheasymptoticexpansionabove,toprovethatthe mappingFμ:m→ffl

Ωθ_m,μ isitselfconvexwheneverμislargeenough.

Remark4.Onecouldalso noticethatthe quantityβ1,m aroseinthe recentpaper[30],where theauthors determinethelargetimebehaviorofadiffusiveLotka-Volterracompetitivesystembetweentwopopulations with growth rates m1 and m2. If β1,m1 > β1,m2, then when μ is large enough, the solution converges as t → +∞ to the steady state solution of ascalar equation associated with the growth rate m1. In other words, thespecieswithgrowth ratem1 chasestheother one.Inthepresentarticle, asabyproduct ofour results,wemaximize thefunctionm→β_1,m. Thisremark impliesthatthisintermediateresultmight find otherapplications ofitsown.

Letusnowformalizerigorouslythereasoningoutlined above,byconsideringanexpansionoftheform θm,μ=m0+η1,m

μ +Rm,μ

μ² . Hence,onehasforallm∈ Mm0,κ(Ω),

d²Fμ(m)[h, h] = 1 μ

ˆ

Ω

¨

η1,m+ 1 μ²

ˆ

Ω

R¨m,μ

Wewill showthatthere holds

d²Fμ(m)[h, h]≥C(h) μ

1−Λ

μ

(22) forallμ>0,whereC(h) andΛ denotesomepositiveconstants.

The strict convexity ofFμ will then follow.Concerning the bang-bang characterof maximizers, notice thattheadmissiblesetMm0,κ(Ω) isconvex,andthatitsextremepointsareexactlythebang-bangfunctions of Mm0,κ(Ω).Oncethe strictconvexityof Fμ showed,wethen easilyinferthatFμ reaches itsmaxima at extreme points, in other words that any maximizer is bang-bang.Indeed, assuming by contradiction the existence of amaximizerwritingtm₁+ (1−t)m₂ with t∈(0,1),m₁ andm₂, two elements of Mm0,κ(Ω) suchthatm1=m2 onapositiveLebesguemeasure set,onehas

Fμ(tm1+ (1−t)m2)< tFμ(m1) + (1−t)Fμ(m2)<max{Fμ(m1),Fμ(m2)}, byconvexityofFμ,whencethecontradiction.

Therestoftheproofisdevotedtotheproofoftheinequality(22).Itisdividedintothefollowingsteps:

Step 1. Uniformestimateof ´

Ωη¨_1,m withrespectto μ.

Step 2. DefinitionandexpansionoftheremaindertermRm,μ. Step 3. Uniformestimateof Rm,μ withrespecttoμ.

Step 1:minoration of´

Ωη¨_1,m Onecomputessuccessively β˙_1,m= 1

m0 Ω

η˙ˆ_1,mm+ ˆη_1,mh

, β¨_1,m= 1 m0

Ω

2 ˙ˆη_1,mh+ ¨η_1,mh

(23)

where η˙ˆ1,m solvestheequation

Δ ˙ˆη_1,m+m₀h= 0 in Ω

∂η˙ˆ1,m

∂ν = 0, on∂Ω with

ˆ

Ω

˙ˆ

η1,m= 0. (24)

Notice moreoverthatη¨_1,m = 0, sinceη˙ˆ1,m islinear withrespect to h.Moreover, multiplyingthe equation abovebyη˙ˆ_1,m andintegrating bypartsyields

β¨1,m = 2 m²₀

Ω

|∇η˙ˆ1,m|²>0 (25)

wheneverh= 0, accordingto (23).Finally,weobtain ˆ

Ω

¨

η1,m =|Ω|β¨1,m+ ˆ

Ω

¨

η_1,m=|Ω|β¨1,m= 2 m²₀

ˆ

Ω

|∇η˙ˆ1,m|².

It isthennotablethat´

Ωη¨_1,m≥0.