Serrin’s type problems in warped product manifolds 2021 Alberto Farina.pdf

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SERRIN’STYPEPROBLEMSINWARPEDPRODUCTMANIFOLDS
ALBERTOFARINAANDALBERTORONCORONI
’soverdeterminedproblemsinwarpedproduct
manifoldsandweproveSerrin’styperigidityresultsbyusingtheP-functionapproachintro-
ducedbyWeinberger.

In[26],:ifthereexistsapositivesolution
u∈C2(Ω)¯tothefollowingsemilinearoverdeterminedproblem

∆u+f(u)=0inΩ,
u=0on∂Ω,(1)

∂νu=con∂Ω,
whereΩ⊂RnisaboundeddomainwithboundaryofclassC2,f∈C1andνdenotesthe
unitnormalto∂Ω,
astheSerrin’ssymmetry(orrigidity)[26]toprovethis
resultisareflnementofthefamousreflectionprincipleduetoAlexandrov(see[1])andisthe
so-calledmovingplanesmethodtogetherwiththemaximumprincipleandanewversionofthe
Hopf’sboundarypointLemma(theCornerlemma,see[26,Lemma1]).Wementionthatthe
techniqueofSerrinappliestomoregenerallyuniformlyellipticoperators(see[26])andithas
inspiredthestudyofvariouspropertiesandsymmetryresultsforpositivesolutionsofelliptic
partialdifferentialequationsinboundedandunboundeddomains(seeforinstancetheseminal
paper[16]).
In[30],∆u=−1basedonwhatare
nowadayscalledP-
ofWeinbergerinspiredseveralworksinthecontextofellipticpartialdifferentialequations(see
.[7,12,13,14,15,23,28]andtheirreferences).
InliteraturetherearegeneralizationsofSerrin’sresultfordomainsintheso-calledspace
forms,,simplyconnectedRiemannianmanifoldswithconstantsectionalcurvature.
ThankstotheKilling-Hopftheorem(see[17,19])itiswell-knownthatspaceformsareisometric
totheEuclideanspaceRn,
[20]and[21]themovingplanesmethodisusedtoprovetheanalogueofSerrin’sresultforthe
arXiv:[]20Dec2019problem(1)forboundeddomainsinHnandinSn(wementionthatinSnthetheoremisnot
+
true,.[11]and[27]).Ofparticularinterestforusarethework[10]and[24]wherea
P-functionapproachisusedtoprovetheanalogueofSerrin’stheoreminspaceformsforthe
followingequation
∆u+nku=−1,(2)
wherek=0inRn,k=1inSnandk=−1inHn(seealso[9]wherethesameproblemiscon-
+
sideredforoverdeterminedproblemsinconvexconesofaspaceform).Inspiredbytheseresults,
inthispaperweprovetheanalogueofSerrin’stheoreminaparticularclassofRiemannian
manifolds:(deflnedflrstlyin[3],seealso
[8])arethemostfruitfulgeneralizationofthenotionofCartesian(ordirect)productandof
Date:December23,2019.
,35N25,53C24(primary);35B50,58J05,58J32(secondary).
,warpedproduct,P-function,rigidity.
1:.
2ALBERTOFARINAANDALBERTORONCORONI
thenotionofrotationallysymmetricmanifold(theoneconsideredin[10]and[24]).Werecall
theirdeflnition
(M,g)ofdimensionn≥2isawarpedproductmanifold
if
M=I×Nandg=dr⊗dr+σ2(r)g,(3)
N
where
•Iisanopeninterval;
•(N,gN)isasmoothandconnected(n−1)-dimensionalRiemannianmanifoldwithout
boundary;
•σ:I→Risasmoothfunctionsuchthat:
σ>0onI.
Inparticular(M,g)isasmoothconnectedn-dimensionalmanifoldwithoutboundary(not
necessarilycomplete).
Ourflrstresultconcernswarpedproductmanifoldswheretheflbermanifold(N,gN)satisfles
RicN≥(n−2)ρgNforsomeconstantρ∈R,(4)
andthefunctionσisgivenby
√√
ce−kr+ce−−krifk<0,
12
σ(r)=c1+c2rifk=0,(5)
√√
c1cos(kr)+c2sin(kr)ifk>0,
wheretheconstantsc1andc2arechosensothatσ>0onI,
σ′(r)≥0forallr∈Iandσ′6≡0(6)
andthewarpedproductmanifold(M,g)satisfles
RicM≥(n−1)kg.(7)
,theRiccitensorofawarped
productmanifoldisgivenbythefollowingexpression
σ′′(r)
Ric=Ric−(σ(r)σ′′(r)+(n−2)σ′(r)2)g−(n−1)dr⊗dr(8)
MNN
σ(r)
(.[4,2])andso
′′′2
σ(r)ρ−σ(r)
RicM=(RicN−(n−2)ρgN)−−(n−2)2g+
σ(r)σ(r)
′′′2
σ(r)ρ−σ(r)
−(n−2)+2dr⊗dr(9)
σ(r)σ(r)
′′′2′′′2
σ(r)ρ−σ(r)σ(r)ρ−σ(r)
≥−−(n−2)2g−(n−2)+2dr⊗dr,
σ(r)σ(r)σ(r)σ(r)
whereinthatletterwehaveused(4).Now,iftheflbermanifold(N,gN)satisfles(4)withρ>0,
theopenintervalIsatisfles(
I⊂(0,∞)ifk≤0,
π(10)
I⊂(0,√)ifk>0,
2k
andthewarpingfunctionisgivenby
√√
sinh(√−kr)
ρ−kifk<0,
√
σ(r)=ρrifk=0,(11)
√√
ρsin(√kr)ifk>0,
k:.
SERRIN’STYPEPROBLEMSINWARPEDPRODUCTMANIFOLDS3
then,thanksto(9),theresultingwarpedproductmanifold(M,g)satisfles(7).Inparticular,
ifN=Sn−1istheunitsphereendowedwithitscanonicalmetric,werecoverthecase(ofopen
sets)ofthespaceformswithconstantsectionalcurvatureequaltok.
Othernon-trivialexamplesofwarpedproductmanifoldssatisfyingourassumptionsareob-
tainedbychoosingNsatisfying(4)andIandσasfollows:
√
I⊂R,σ(r)=e−kr,k<0andρ=0,(12)
or
√√
I⊂(0,∞),σ(r)=ce−kr+ce−−kr,k<0,c≥c>0suchthatρ=4kcc(<0).(13)
121212
Alsoobservethat,when(11)or(12)or(13)areinforceandtheflbermanifold(N,gN)is
Einstein(.,itsatisfles(4)withtheequalitysign)thenalsotheresultingwarpedproduct
manifold(M,g)isEinstein(.,itsatisfles(7)withtheequalitysign).
Withthesepreliminaries,theflrstresultofthispaperisthefollowing.
(M,g)beawarpedproductmanifold(notnecessarilycomplete)suchthat(4),
(5),(6)and(7)⊂Mbeadomain()withboundary
∈C3(Ω)∩C2(Ω)beasolutionto

∆u+nku=−1inΩ,
u>0inΩ,(14)

u=0on∂Ω,
suchthat,forsomeconstantc,
|∇u|=con∂Ω.(15)
ThenΩisametricballanduisaradialfunction,
centeroftheball.
ThesecondresultofthispaperisageneralizationofTheorem2ingeneralwarpedproduct
manifolds,.,forgeneralwarpingfunctionsσ>0,whereweassumeacompatibilitycondition
betweenthegeometryofthewarpedproductandthesolutiontotheoverdeterminedproblem.
(M,g)beawarpedproductmanifold(notnecessarilycomplete)suchthat
RicM≥(n−1)kg,forsomek∈R,(16)
and
σ′(r)≥0forallr∈I,σ′6≡0.(17)
LetΩ⊂∈C3(Ω)∩C2(Ω)
beasolutionto(14)-(15).Ifusatisflesthefollowingcompatibilitycondition
Z′′n−1′
′(σσ)2
kσ+n−1u≥0.(18)
Ωnσ
ThenΩisametricballanduisaradialfunction,
centeroftheball.
ObservethatinTheorem3wedonotassume(4)andobservethatifweassumethatσis
givenby(5),thenconditions(16)and(17)aretriviallysatisfled,moreoveralsothecompatibility
conditionistriviallysatisfled.
Thisresultimprovestheresultin[24]wherethecaseofmodelmanifoldsandk=0was
,in[24]theproblemis∆u=−1onmodelmanifoldswithnonnegative
Riccicurvatureandsuchthatσ′>0,moreoverthereisacompatibilityconditionwhichisexactly
(18)withk=0(see[24,Formula3]).Theconclusionin[24]isstrongerthantheoneinTheorem
3,indeedin[24]
strongrigidityresultwillbegeneralizedinournextresult(seeTheorem4inthesequel).To:.
4ALBERTOFARINAANDALBERTORONCORONI
thisendweobservethatthemodelmanifoldscanbewrittenaswarpedproductmanifoldswith
apole,explicitlyonetakes:
I=[0,R)withR≤+∞andN=Sn−1(endowedwithitscanonicalmetric),
togetherwiththerighthypothesisonthefunctionσwhichmakesthemetricgin(1)smooth
(see[24,Deflnition2]).
modelmanifoldsandprovidesafullgeneralizationof[24,Theorem3].Inparticular,underthe
assumptionthatthepoleofthemodeloisinsidethedomain,weprovethatthedomainmust
beageodesicballaroundoandthatthemetricintheballhasconstantsectionalcurvatures.
(M,g)beamodelmanifold(notnecessarilycomplete)suchthat(16)and
(17)⊂MbeadomainwithboundaryofclassC1suchthatΩiscompact.
Weassumethato∈∈C3(Ω)∩C2(Ω)beasolutionto(14)-(15).Ifusatisflesthe
compatibilitycondition(18).ThenΩisametricballcentredatoofradiusρanduisaradial
functiongivenby
√
cosh(√−kr)−1ifk<0,
kncosh(−kρ)nk
ρ2r2
u(r)=2n−√2nifk=0,(19)
cos(kr)
√−1ifk>0,
kncos(kρ)nk
whereristhegeodesicdistancefromo.
Moreover,thewarpingfunctionσinthemetricballisgivenbythefollowingexpression:
√
sinh(√−kr)ifk<0,
−k
σ(r)=rifk=0,(20)
√
sin(√kr)ifk>0.
k
ObservethatinTheorem4ifweassumethatσisgivenby(20)then(16),(17)andthe
compatibilitycondition(18)areautomaticallysatisfled.
BeforewegettotheheartofthepaperwecommentaboutourTheorems.
Theorems2and4recoverandimprovetheresultin[10]wherethedomainΩwasabounded
domainofoneofthefollowingthreemodels:theHyperbolicspaceHn,theEuclideanspaceRn
,thesemanifoldsaremodelmanifolds(takeI=[0,+∞)forHn
+
andRn,I=[0,π)forSnandσasin(20),withk∈{−1,0,1})andsowerecovertheresultsin
2+
[10]byapplyingTheorem4,ifthepolebelongstoΩ,andbyapplyingTheorem2whenthepole
doesnotbelongtoΩ(inthelatterwehaveseenΩasaboundeddomainofthewarpedproduct
manifolds:Hnminusonepoint,RnminusonepointandSnminusonepoint(thepole)).
+
Moreover,theauthorsof[10]donotrequirethatthesolutionof(14)ispositive,indeedin
thecasesofRnandofHnthisfollowsfromthestandardmaximumprinciples,whileinthe
caseofSnthisfollowsfromthefactthattheflrsteigenvalueoftheDirichletLaplacianonthe
+
hemisphereisnandthecorrespondingeigenfunctionisstrictlypositive(see[10,ProofofLemma
]fordetails).Thisprovesthatthemainresultin[10]isaspecialcaseofourTheorem2and
Theorem4.
=
(a,b+ε),with0<a<b<∞,ε>0,Nsuchthat(4)holdswithρ≥1and
(
rin(a,b],
σ(r)=−1
r(1−er−b)in(b,b+ε).
Themetricgivenbydr⊗dr+σ2(r)gNissmoothandthecorrespondingwarpedproductisa
noncompleteRiemannianmanifold(arotationallysymmetricsmoothmanifoldifN=Sn−1is
theunitsphereendowedwithitscanonicalmetric).Moreoverthewarpedproductsatisflesthe
hypothesisofTheorem3,indeedfromadirectcomputationbymakinguseofformula(8)we:.
SERRIN’STYPEPROBLEMSINWARPEDPRODUCTMANIFOLDS5
have,forεsufficientlysmall,
(
RicM=0foranypointin(a,b]×N,
(21)
RicM≥0foranypointin(b,b+ε)×N,
hence(16)holdswithk=0and(17),theideaistotakethedomain
Ωin(a,b)×N,wherethemetricistheEuclideanone,andsoalso(18),
summingup,wehavethatthisisannontrivialexampleinwhichourTheorem3works.
ThisexamplealsoappliesinthecaseofTheorem4(justconsiderthemodelmanifoldobtained
bytakingI=[0,b+ε)).
TheproofofallresultsisbasedonthefollowingP-function(consideredalsoin[30]in[10]
andin[24])
222
P(u):=|∇u|+u+ku,(22)
n
whereuisthesolutionto(14)-(15).
:inSection2weprovegeneral
resultsrelatedtowarpedproductmanifolds,inSection3weprovethemainresultsofthepaper
andinAppendixAweproveageneralpropertyofaboutthestar-shapednessofgeodesicballs
insideageneralRiemannianmanifold.

’AnalisiMatem-
atica,laProbabilit´aeleloroApplicazioni(GNAMPA)oftheIstitutoNazionalediAltaMatem-
atica(INdAM).,Universit´e
dePicardieJulesVerneinAmiens,whichisacknowledgedforthehospitality.


-
-typeidentitywhichisthefundamentalingredientsintheproofsofall
Theorems.
--function(22)issubhar-
monicaccrodinglytothefollowinglemma,moreoverweareabletocharacterizetheharmonicity
ofP;thisisaverygeneralresultwhichholdstrueineveryRiemannianmanifoldwithRicci
curvatureboundedfrombelow.
(M,g)beann-dimensionalRiemannianmanifold(notnecessarilycomplete)
suchthat
RicM≥(n−1)kgfork∈R.(23)
LetΩ⊂Mbeadomainandletu∈C2(Ω)beasolutionto
∆u+nku=−1inΩ.(24)
Then
∆P(u)≥0inΩ.
wherePisgivenby(22).Moreover,
∆P(u)=0