Solutions to theσk-Loewner-Nirenberg Problem on Annuli are Locally Lipschitz and Not Differentiable

YanYan Li and Luc Nguyen

1 Department of Mathematics,Rutgers University,Hill Center,Busch Campus,110 Frelinghuysen Road,Piscataway,NJ08854,USA;

2 Mathematical Instituteand St Edmund Hall,University of Oxford,Andrew Wiles Building,Woodstock Road,Oxford OX2 6GG,UK.

Abstract.We show for k≥2 that the locally Lipschitz viscosity solution to theσk-Loew ner-Nirenberg problem on a given annulus{a<|x|

Key w ords:σk-Loew ner-Nirenberg problem,σk-Yamabe problem,viscosity solution,regularity,conformal invariance.

1 Introduction

Let?be a smooth bounded domain in Rn,n≥3.For a positiveC2functionudef ined on an open subset of Rn,letAudenote its conformal Hessian,namely

For 1≤k≤n,letσk:Rn→R denotek-th elementary symmetric function

and letΓkdenote the coneΓk={λ=(λ1,...,λn):σ1(λ)>0,···,σk(λ)>0}.

In[7,Theorem 1.1],it was shown that theσk-Loewner-Nirenberg problem

Eq.(1.2)is a fully nonlinear elliptic equation of the kind considered by Caffarelli,Nirenberg and Spruck[3].We recall the following def inition of viscosity solutions which follows Li[20,Def initions 1.1 and 1.1’](see also[19])where viscosity solutions were f irst considered in the study of nonlinear Yamabe problems.

Let

Def inition 1.1.Let??Rn bean open set and1≤k≤n.Wesay that an upper semi-continuous(alower semi-continuous)function u:?→(0,∞)isasub-solution(super-solution)to(1.2)in the viscosity sense,if for any x0∈?,?∈C2(?)satisfying(u??)(x0)=0and u??≤0(u??≥0)near x0,thereholds

Wesay that a positivefunction u∈C0(?)satisf ies(1.2)in theviscosity senseif it is both a sub-and asuper-solution to(1.2)in theviscosity sense.

Eq.(1.2)satisf ies the following comparison principle,which is a consequence of the principle of propagation of touching points[23,Theorem 3.2]:Ifvandware viscosity sub-solution and super-solution of(1.2)and ifv≤wnear??,thenv≤win?;see[7,Proposition 2.2].The above mentioned uniqueness result for(1.2)-(1.3)is a consequence of this comparison principle and the boundary estimate(1.4).

In the rest of this introduction,we assume that?is an annulus{a<|x|

Our f irst result improves on the above non-existence ofC2solutions to(1.2)-(1.3).

We observe the following result,which is essentially due to Gursky and Viaclovsky[12].We provide in the appendix the detail for the piece which is not directly available from[12].

Theorem 1.3.Supposethat n≥3,2≤k≤n,and(Mn,g)is a compact Riemannian manifold.If λ(?Ag)∈Γk on M,then(1.7)hasa Lipschitz viscosity solution.

Here viscosity solution is def ined analogously as in Def inition 1.1.

Wesay that a positive function u∈C0(M)satisf ies(1.7)in theviscosity senseif it is both a sub-and asuper-solution to(1.7)in theviscosity sense.

In both contexts,it is an interesting open problem to understand relevant conditions on?,or on(M,g),which would ensure that(1.2)-(1.3),or(1.7)respectively,admits a smooth solution.We make the following conjecture.

Conjecture 1.1.Suppose thatn≥3,2≤k≤n,and??Rnis a bounded smooth domain.Then the locally Lipschitz viscosity solution to(1.2)-(1.3)is smooth near??.

Some further questions are in order.

Question 1.1.Suppose thatn≥3,2≤k≤n,and??Rnis a bounded smooth domain.If(1.2)-(1.3)has a smooth sub-solution,must(1.2)-(1.3)have a smooth solution?

Question 1.2.Suppose thatn≥3,2≤k≤n,and??Rnis a smooth strictly convex(non-empty)domain.Is the locally Lipschitz viscosity solution to(1.2)-(1.3)smooth?

If?is a ball,then thesolution to(1.2)-(1.3)issmooth and correspondsto the Poincar′e metric.

Question 1.4.Suppose thatn≥3,2≤k≤n,and(Mn,g)is a Riemannian manifold such thatλ(?Ag)∈ΓkonM.Does(1.7)have a unique Lipschitz viscosity solution?

It is clear that(1.7)has at most oneC2solution by the maximum principle.In fact,if(1.7)has aC2solution,then that solution is also the unique continuous viscosity solution in view of the strong maximum principle[2,Theorem 3.1].Equivalently,if(1.7)has two viscosity solutions,then it has noC2solution.

Question 1.5.Suppose thatn≥3 and 2≤k≤n.Does there exist a Riemannian manifold(Mn,g)such thatλ(?Ag)∈ΓkonMand(1.7)has a Lipschitz viscosity solution which is notC2?

Finally,we discussthe case where(1.3)is replaced by f inite constant boundary conditions

Theorem 1.4.Supposethat n≥3and2≤k≤n.Let?={a<|x|

(i)u is smooth in each of{a≤|x|

(iii)and thef irst radial derivative?ru jumpsacross{|x|=m}:

Remark 1.1.It is clear from Theorem 1.4(in Cases 1–3)that ifuis aC1and radially symmetric solution to(1.2)in theviscosity sensein someopen annulus?thenu∈C∞(?).

Remark 1.2.In Case 4,theexact value ofmis

wherepis the solution to

The following question is related to Question 1.

For comparison,we recall here a result of Bo Guan[8]on the Dirichletσk-Yamabe problem in the so-called positive case which states that the existence of a smooth subsolution implies the existenceof a smooth solution.

We concludethe introduction with one more question.

Question 1.7.Letn≥3,2≤k≤nandm/=n?1.Does there exist a smooth domain??Rnsuch that the locally Lipschitz solution to(1.2)-(1.3)isC2away from a setΣwhich has Hausdorff dimensionm?

In Section 2,we prove all the results above except Theorem 1.3,whose proof is done in the appendix.Theorem 1.1 is proved f irst in Subsection 2.1.We then prove a lemma on the existence and uniqueness a non-standard boundary value problem for the ODE related to(1.2)in Subsection 2.3 and use it to prove Theorem 1.2 in Subsection 2.4 and Theorem 1.4 in Subsection 2.5.

2 Proofs

2.1 Proof of Theorem 1.1

We will usethe following lemma.

Mij(δij?mimj)≥0.

the conclusion follows.

Proof.Fix somep∈Σn?1and letνbe a unit vector atf(p)normal to the image of a small neighborhood ofp,.Recall that

This means

Using(2.1)yields the conclusion.

Integrating overΣ,we thus have thatHΣ≡0 and?u≡const onΣ.In particular,f:Σn?1→?is a minimal immersion with respect to?g.This is impossible as there is no smooth minimal immersion in Rnwith codimension one.

2.2 Preliminary ODEanalysis

By the uniqueness result in[7,23],the solutionsuin Theorems 1.2 and 1.4 are radially symmetric,u(x)=u(r)wherer=|x|.

As in[5,32],wework on a round cylinder instead of Rn.Namely,let

where here and below′denotes differentiation with respect tot.

Note that,fork≥2,at points whereuis twice differentiable,λ(?Au)∈Γkif and only ifσk(λ(?Au))>0 and|ξ′|>1.Indeed,ifσk(λ(?Au))>0 and|ξ′|>1,then(2.2)impliesσi(λ(?Au))>0for 1≤i≤kand soλ(?Au)∈Γk.Conversely,ifλ(?Au)∈Γkfor somek≥2,then

Using(2.2),we seethat the f irst two inequalities imply|ξ′|>1.

We are thus led to study the differential equation

under the constraint that|ξ′|>1.

It is well known(see[5,32])that(2.3)has a f irst integral,namely

A plot of the contours ofHfork=2,n=7 is provided in Figure 1.See[5]for a more complete catalog.

Before moving on with the proofs of our results,we note the following statement.

Remark 2.1.As a consequence of Theorem 1.4,we have in fact thatH(ξ,ξ′)is(locally)constant along viscosity solutions.

Figure 1:The contours of H for k=2,n=7.Each radially symmetric viscosity solution to(1.2)lies on a single contour of H but avoid the shaded region,i.e.the dotted parts of the contours of H are excluded.Every smooth solution stays on one side of the shaded region.Every non-smooth solution jumps(on one contour)from the part below the shaded region to the part above the shaded region at a single non-differentiable point.

HenceH(ξ,ξ′)is also constant in{?a

2.3 A lemma

Proof.We use ideas from[5].

By(2.6)we thus have

In this proof,wewill only need to consider thecasethat(?1)k H(p,q)<0.Then by[5](Theorem 1,Case II.2 for evenkand Theorem 2,Case II.2 for oddk),we have thatTp,qis also f inite(corresponding tor?being f inite in the notation of[5])and

Before moving on to the next stage,we note that,in view of(2.6),

In particular,then length ofIp,qdependsonly onn,kand thevalueofH(p,q),rather thanpandqthemselves.

2.4 Proof of Theorem 1.2

solves(1.2)-(1.3)in{a

(i)limt→±Tξ(t)=?∞,

(ii)ξ′(0?)=1,ξ′(0+)=?1,

(iii)and|ξ′|>1 in(?T,0)∪(0,T).Indeed,letξT:[0,T)→R be the solution obtained in Lemma 2.3,and def ine

there is noC2function?such that?≥unearx0and?(x0)=u(x0).Therefore(a)holds.

Suppose now that?is aC2function such that?≤unearx0and?(x0)=u(x0).Asuis radial,this implies that

LetOdenote the diagonal matrix with diagonal entries 1,?1,...,?1.Note that,in block form,

Thus,by(2.18),

Also,?(x0)=ˉ?(x0)and,in view of(2.17),??(x0)=?ˉ?(x0).Hence

2.5 Proof of Theorem 1.4

(i)Suppose thatT

(ii)SupposethatT=T(a,b,c1,c2).We show that Case 3 holds.

(iii)Suppose thatT>T(a,b,c1,c2).We show that Case 4 holds.

In this case,we selectp≥pa(≥pb)such that

Suchpexists as the right hand side tends toT(a,b,c1,c2)whenp→paand diverges to∞asp→∞.Recall the solutionξpdef ined in the proof of Lemma 2.3.Let

Then 2T±

We then let

Wecan then proceed as in theproof of Theorem 1.2 to show thatξis thedesired solution.

□

AAppendix:Proof of Theorem 1.3

By[12,Theorem 1.4],we have for all suff iciently smallτ>0 that the problem

has a unique smooth solutionuτ.Furthermore,by[12,Propositions 3.2 and 4.1],the family{uτ}is bounded inC1(M)asτ→0.(C2bounds foruτwere also proved in[12],but these bounds are unbounded asτ→0.)Hence,along some sequenceτi→0,uτiconverges uniformly to someu∈C0,1(M).To conclude,we show thatuis a viscosity solution to(1.7).

For notational convenience,we renameuτiasui.Fix someˉx∈M.

Step 1:We show thatuis a sub-solution to(1.7)atˉx.More precisely,we show that for every?∈C2(M)such that?≥uonMand?(ˉx)=u(ˉx)there holds that

Note that

and so

Recalling(A.1),we hence have

Sinceδis arbitrary,this proves(A.2)after sendingδ→0.

Step 2:We show thatuis a super-solution to(1.7)atˉx,i.e.if?∈C2(M)is such that?≤uonMand?(ˉx)=u(ˉx),then

By(A.1),we hence have

Acknowledgment

The authors would like to thank Matt Gursky and Zheng-Chao Han for stimulating discussions.Theauthorsare grateful to therefereesfor their very careful reading and useful comments.YanYan Liis partially supported by NSFGrant No.DMS-1501004.