DSpace at VNU: Existence Results for the Einstein-Scalar Field Lichnerowicz Equations on Compact Riemannian Manifolds in the Null Case

DSpace at VNU: Existence Results for the Einstein-Scalar Field Lichnerowicz Equations on Compact Riemannian Manifolds in...

Digital Object Identifier (DOI) 10.1007/s00220-014-2133-7 Mathematical

Physics

Existence Results for the Einstein-Scalar Field

Lichnerowicz Equations on Compact Riemannian

Manifolds in the Null Case

Qu´ôc Anh Ngô1,2, Xingwang Xu3

1 Laboratoire de Mathématiques et de Physique Théorique (LMPT), CNRS-UMR 7350,

Université François Rabelais de Tours, Parc de Grandmont, 37200 Tours, France.

E-mail: quoc-anh.ngo@lmpt.univ-tours.fr

2 Department of Mathematics, College of Science, Vietnam National University, Hanoi, Vietnam E-mail: bookworm_vn@yahoo.com

3 Department of Mathematics, National University of Singapore, Block S17 (SOC1), 10 Lower Kent Ridge Road, Singapore 119076, Singapore E-mail: matxuxw@nus.edu.sg

Received: 6 October 2013 / Accepted: 15 April 2014

© Springer-Verlag Berlin Heidelberg 2014

Abstract: This is the second in our series of papers concerning positive solutions of

the Einstein-scalar field Lichnerowicz equations Let(M, g) be a smooth compact

Rie-mannian manifold without boundary of dimension n  3, f and a  0 are two smooth

functions on M with

M f dvg < 0, sup M f > 0, andM a dvg > 0 In this article,

we prove two results involving the following equation arising from the Hamiltonian constraint equation for the Einstein-scalar field equation in general relativity

 gu = f u2−1+ a

u2+1,

where  g = − divg (∇ g ) and 2  = 2n/(n − 2) First, we prove that if either sup M f

and

M a dvgor supM a is sufficiently small, the equation admits one positive smooth

solution Second, we show that the equation always admits one and only one positive smooth solution provided supM f  0 We should emphasize that we allow a to

van-ish somewhere Along with these two results, existence and non-existence for related equations are also considered

Contents

1 Introduction

2 Preliminary

2.1 Notations and basic properties forλ f andλ f ,η,q

2.2 Basic properties for positive solutions

2.3 A necessary condition for f

2.4 The non-existence of smooth positive solutions of suitable small energy 3 The Analysis of the Energy Functionals When supM f > 0

3.1 Functional setting

3.2 Asymptotic behavior ofμ ε k ,q

4 Proof of Theorem 1.1

4.1 The case infM a > 0

4.2 The case infM a= 0

5 Proof of Theorems 1.2 and 1.3

5.1 Proof of Theorems 1.2

5.2 Proof of Theorem 1.3

6 Some Remarks

6.1 Construction of functions satisfying (1.9)

6.2 A relation between supM f and M a dv g

6.3 A stability result for small|h|

6.4 Proof of Lemma 2.1

References

1 Introduction

This is the second in our series of papers concerning positive solutions of the Einstein-scalar field Lichnerowicz equations (ELEs for short) on compact Riemannian manifolds Roughly speaking, given a smooth compact Riemannian manifold(M, g) without the

boundary of dimension n 3, the ELEs can be written as the following simple partial

differential equation

 gu + hu = f u2−1+ a

u2+1, u > 0, (1.1) where  g = −divg (∇ g ) is the Laplace-Beltrami operator, 2  = 2n/(n − 2) is the

critical Sobolev exponent, and h , f, a (0) are smooth functions in M In the literature,

Eq (1.1) is motivated by the Hamiltonian constraint equations naturally arising when solving the Cauchy problem in general relativity through the conformal method Due to the nature of their origin, Eq (1.1) has recently received much considerable attention from the mathematical analysis point of view, for example [7,11,16–20]

For the sake of clarity and in order to make the paper self-contained, let us briefly recall how the conformal method can be used when the Cauchy problem in general relativity is studied and how we come up with (1.1) Mathematically, for a given initial data set(M, g, K ) consisting of an n-dimensional Riemannian manifold (M, g) and a

symmetric(0, 2)-tensor K , the initial value problem asks for a Cauchy development of

(M, g, K ), denoted by (V, g), which is a Lorentzian manifold of dimension n + 1 Here

the spacetime metric g is required to satisfy the following Einstein equation

Ricg−1

2Scalg g = T,

where Ricg and Scalg are the Ricci tensor and the scalar curvature of the spacetime

metric g Also, the symmetric (0, 2)-tensor T appearing in the Einstein equation is the

energy-momentum tensor which is supposed to present the density of all the energies, momenta and stresses of the sources, see [4, Chapter III]

In order for (V, g) to be a Cauchy development of (M, g, K ), it is required that (M, g, K ) must embed isometrically to (V, g) as a slice with the second fundamental

form K ; and the metric g becomes the pullback of the spacetime metric g by the

embed-ding It turns out that the initial data(g, K ) cannot be arbitrary, they must satisfy some

conditions In view of the Gauss and Codazzi equations, those conditions can be rewrit-ten in a form consisting of two equations known as the Hamiltonian and momentum constraints defined on(M, g) as shown below

Scalg−|K |2

g+(tr g K )2= 2ρ,

where all quantities of (1.2) involving a metric are computed with respect to g and Scalg

is the scalar curvature of g Also in (1.2),ρ is a scalar field on M representing the

energy density and J is a vector field on M representing the momentum density of the nongravitational fields; they are related to the energy-momentum tensor T as follows

ρ = T (n, n), J = −T (n, ·),

where n is the unit timelike normal to the slice M× {0}, see [4,5] and [6, Section5]

It follows from a simple dimension counting argument that the constraint equations in(1.2) form an under-determined system; thus they are in general hard to solve However, itwas remarked in [4] that the conformal method can be effectively applied in the constantmean curvature setting, that is to look for(g, K ) of the following form

where g is fixed, u is a positive (smooth) function, and W is a 1-form Note that the

operatorL appearing in (1.3) is the conformal Killing operator acting on W defined in

local coordinates by

LW i j = ∇i W j +∇j Wi−2

n (∇ k Wk )g i j ,

where ∇ and ∇ are the Levi-Civita connections associated to the metrics g and g

respectively Here byτ = g i j K i j , we mean the mean curvature of M as a slide of V

The choice forσ is somehow arbitrary; however, it is related to the York splitting The

novelty of using the decomposition (1.3) is that the system (1.2) is easily transformed

to a new determined system of partial differential equations of variable(u, W) given as

Whenτ is constant, Eq (1.4b) then only involves W and generically implies W ≡ 0

(for example, if M admits no conformal Killing vector field) Therefore, one is left with

solving Eq (1.4a) In the vacuum case, e.g T ≡ 0 and hence ρ ≡ 0 and J ≡ 0 as

well, we know exactly which sets of data lead to solutions and which do not, see [12].However, in the non-vacuum case, it should be pointed out that there are several casesfor which either partial result or no result was achieved when solving (1.4a), especiallywhen gravity is coupled to scalar field sources To see this more precisely, we assumethe presence of a real scalar fieldψ in the space time (V, g) with a potential U being a

function ofψ The energy-momentum tensor T of a real scalar field is then given by

Ti j = ∇ i ψ ∇ j ψ −1

2g i j ∇ k ψ ∇ k ψ −g i j U (ψ).

A direct computation then leads us to

2ρ = π2+|∇ψ|2

g + 2U (ψ),

whereπ is the normalized time derivative of ψ restricted to M and ψ is the restriction

ofψ to M, see [4,5] for details As already shown in [5], to avoid introducing any newvariable, the only way to decompose the scalar field(ψ, π) is the following

then we easily verify that (1.1) is nothing but (1.7) Based on a division recently obtained

in [5], one can see that when solving (1.1) there are two cases corresponding to either

h < 0 or h ≡ 0 with sign-changing f , for which no result was achieved This is basically

due to the fact that the method of sub- and super-solutions does not work, thus forcing

us to develop a new approach

In the preceding paper [17], we have already proved that, in the case h < 0, a suitable

balance between coefficients h, f , a of (1.1) is enough to guarantee the existence of onepositive smooth solution In addition, it was found that under some further conditions

we may or we may not have the uniqueness property for solutions of (1.1) This paper

is a continuation of the paper [17] To be precise, in the present paper, we continue ourstudy of the non-existence and the existence of positive smooth solutions to (1.1) when

h = 0 which was also left as an open question in the classification of [5], that is, we areinterested in the following simple partial differential equation

We assume hereafter that f and a are smooth functions on M with a  0 The latter

assumption implies no physical restrictions since we always have a 0 in the original

Einstein-scalar field theory Besides, in order to avoid studying the same equation arisingfrom the prescribing scalar curvature problem in the null case, see [8], it is natural toassume

M a dvg > 0 Thanks to the conformally covariance property of (1.2), we can

freely choose a background metric g such that the manifold M has unit volume.

In the first part of the present paper, we mainly consider the case supM f > 0 and



M f dvg < 0 (this is also a necessary condition if a ≡ 0) Before stating the result, let

us denote by f±the positive and negative parts of f , i.e., we define f−= min{ f, 0}  0

and f+ = max{ f, 0}  0 Using these notations, we are able to show that if sup M f+

and

M a dvg are bounded from above by constants depending on n and f−, then (1.8)

possesses at least one smooth positive solution Following is the statement:

Theorem 1.1 Let (M, g) be a smooth compact Riemannian manifold without the

bound-ary of dimension n  3 Assume that f and a  0 are smooth functions on M such that



M a dvg > 0,M f dvg < 0, and sup M f > 0 Then there exist two positive constants

C1and C2to be specified such that if

hold, then (1.8) possesses at least one smooth positive solution.

To be precise, the constants C1and C2appearing in Theorem1.1are given in (4.1)–(4.2) below The question of whether we can find an explicit formula for C1and C2

turns out to be difficult, even for the prescribed scalar curvature equation, for interestedreaders, we refer to [2]

Combining with [17, Theorem 1.1], it turns out that existence result for the cases h=

0 and h < 0 are in a similar fashion However, as already seen in the case h > 0 where

we are able to keep either (1.9) or (1.10) and drop the other condition, the requirement

for the case h = 0 cannot be as strong as that in the case h < 0 Surprisingly enough,

in the next result of the present paper, we would like to emphasize that if we replace the

estimate for L1-norm of a in (1.10) by a suitable estimate for L∞-norm of a, then the

condition (1.9) can be dropped The proof we provide here is based on the method ofsub- and super-solutions, see [13,15]

Theorem 1.2 Let (M, g) be a smooth compact Riemannian manifold without the

bound-ary of dimension n  3 Assume that f and a  0 are smooth functions on M such that



M a dvg > 0,M f dvg < 0, and sup M f > 0 Then there exists a positive constant C3

depending only on f and n such that if

sup

M

then (1.8) possesses at least one smooth positive solution.

Again, the constant C3appearing in the theorem above which is given in (5.1) below

is less explicit Concerning (1.1), using previous results for the negative case in [17] andfor the positive case in [10,18] together with Theorems1.1–1.2above, one can obtain

in the case when f changes sign a picture of the interaction between the coefficients of

(1.1) when h varies from −λ f to +∞ in order for (1.1) to get solutions, see Table1fordetails

In the last part of the present paper, we focus our attention to the case supM f  0

We shall prove the following result

Theorem 1.3 Let (M, g) be a smooth compact Riemannian manifold without boundary

of dimension n  3 Let f and a be smooth functions on M with a  0 in M,M a dvg >

0, and f  0 Then (1.8) possesses a unique positive solution.

Concerning Theorem 1.3, it is worth noticing that it generalizes the same resultobtained in [5] where the method of sub- and super-solutions was used In this paper, weprovide a variational approach to prove this result As can be seen in both theorems, we

allow a to have zeros in M In order to achieve this goal, we make use of the sub- and

super-solution method as suggested in [9] While the existence of a super-solution is quiteeasy to see, a sub-solution is rather hard to construct We believe that our construction

of a sub-solution could be useful elsewhere

Before closing this section, let us briefly mention the organization of the paper andhighlight some techniques used In Sect 2, we discuss the quantities λ f andλ f ,η,q concerning the positive part f+ of f as well as basic properties of positive solutions

of (1.8) Also in this section, we prove that the condition

M f dvg < 0 is necessary.

In Sect.3, a careful analysis of the energy functional under the case supM f > 0 is

presented Having these preparations, we spend Sect 4to prove Theorems1.1whileTheorems1.2and1.3will be proved in Sect.5 In Sect.6, we provide a procedure to

construct a function f which satisfies (1.9) In addition, we also comment on the relationbetween supM f and

holds For simplicity, we denote by 2 the average of 2 and 2, that is, 2 = (2n −

2)/(n − 2) Following [21], we define the following number

Intuitively, functions inA can be thought of as functions that vanish on the support of f−.

As can be seen from [17], the numberλ f plays an important role when solving Eq (1.1)

in the negative case, namely h < 0, see also [21] We recall from [17, Proposition 2.5]that in the negative case, it is necessary to have λ f > |h| in order for (1.1) to havepositive solutions In addition, the strict inequalityλ f > |h| is important for arguments

used in [17] Then, it is naturally to expect that the following conditionλ f > 0 should

hold in the null case

Up to this point, it is not clear whether or not the strict inequalityλ f > 0 actually

holds in the null case This is because the method used in [17, Proposition 2.5] does not

work for the case h = 0 However, in view of [21, Lemme 1], the numberλ f, if finite,coincides with the first positive eigenvalueλ1of the associated Dirichlet problem over

the region M1= {x ∈ M : f (x)  0} Hence, we obtain the following result.

Lemma 2.1 There holds λ f > 0.

Surprisingly, although we cannot directly adopt the method used in [17] to proveLemma2.1above, a small change to Eq (1.1) simply by adding the term−u/n to the

left hand side of (1.1) leads to a proof for Lemma2.1 For the sake of completeness, weprovide this proof in Subsection6.4at the end of the paper

In the following, we approximateλ f by usingλ f ,η,q as proposed in [21] For each

η ∈ (0, 1) fixed, we let A (η, q) be a subset of H1(M) defined as

well-by some positive functionv ∈ A (η, q) We now mention one useful property of λ f ,η,q

whose proof can be found in [17,21] For the sake of clarity, we divide the statement ofthat property into the following two lemmas

Lemma 2.2 Suppose λ f < +∞ For each δ > 0 fixed, there exists η0 > 0 such that for

all η < η0 , there exists q η ∈ (2 , 2  ) so that λ f ,η,q  λ f − δ for every q ∈ (q η , 2  ).

Lemma 2.3 Suppose λ f = +∞ There exists η0 > 0 such that for all η < η0 , there exists q η ∈ (2 , 2  ) so that λ f ,η,q  1 for every q ∈ (q η , 2  ).

Having the numberλ f, we then introduce the following quantity

K1+ 2A1/λ f −1, if λ f < +∞,

1

In view of Lemmas2.2and2.3, there exist two numbersη0 ∈ (0, 1) and q η0 ∈ (2 , 2  )

so that the following estimate

λ f ,η0,q 



λ f /2, if λ f < +∞,

holds for every q ∈ (q η0, 2  ) In addition, since λ f ,η,q is monotone decreasing inη, see

[17,21], we may assume in the caseλ f = +∞ that

since we may take η0 as small as we wish It is important to note that, in the casesupM f > 0, the number η0 depends only on the negative part f−of f Unless otherwise

stated, from now on, we fix such anη0 and we only consider q ∈ [q η0, 2  ) Finally, we

introduce the following numbers

k1,q = η0

2q



λq η0M | f−| dv g

q /(q−2)





λq η0M | f−| dv g

q /(q−2)

From the choice ofη0 , one can see that k1,q < k2 ,q for any q ∈ [2 , 2  ) One can easily

bound k1,q from below and k2,q from above, to be precise, there exists two positive

numbers k < 1 and k > 1 independent of q and ε such that k  k1 ,q < k2 ,q  k For

example, since 2 /(2 − 2) = n − 1 one can choose

n−1

, 1



2.2 Basic properties for positive solutions As already used in [17] for the case h < 0,

the original idea of our approach was based on a mini-max method in a paper by Rauzy[21] However, we find that in the case considered in [21], the assumption of the negativeYamabe invariant is important; in fact, his approach does not work for the case of the nullYamabe invariant in the prescribing scalar curvature problem Moreover, the standard

sub- and super-solutions method also does not work either since f changes sign As a

first step to tackle (1.8), we look for positive smooth solutions of the following subcriticalequation

which does include (1.8) as a particular case We spend this subsection studying severalproperties of positive solutions of (2.8) First, we derive a lower bound for a positive C2solution u of (2.8) which is independent of q and ε.

Lemma 2.4 Let u be a positive C2solution of (2.8) Then there holds



Proof Let us assume that u achieves its minimum value at x0 For simplicity, let us

denote u (x0), f (x0), and a(x0) by u0 , f0, and a0respectively Notice that u0> 0 since

u is a positive solution We then have  gu|x0  0 and

f0(u0)q−1+ a0u0

((u0)2

Consequently, we get that f0< 0 Using (2.11) we can see that

a0 − f0(u0)q−2((u0)2+ε) q /2+1  − f0((u0) 2+ε) q

which implies that

Thus, one can conclude that u0satisfies (2.9) for any q ∈ [2 , 2  ) and any ε verifying

the condition (2.10) The proof is complete

As can be seen from the proof above, although our lower bound is independent of q

andε, it depends on inf M a A recent attempt due to Premoselli suggests that, in the case

{u q}qof (2.8) as q 2regardless of infM a For the interested reader, we refer to [20,

Proposition 3.1] We now quote the following regularity whose proof can be mimickedfrom a similar result proved in [17]

Lemma 2.5 Assume that u ∈ H1(M) is an almost everywhere non-negative weak

solution of Eq (2.8) We assume further that inf M a > 0 Then

(a) If ε > 0, then u ∈ C∞(M) In particular, u  0 in M.

(b) If ε = 0 and u−1∈ L p (M) for all p  1, then u ∈ C∞(M).

It is worth mentioning that there is an extra assumption in Lemma2.5above compared

to [17, Lemma 2.2] To be precise, we require infM a > 0 in Lemma2.5and it seemsthat this assumption is just a technical assumption The reason is that we need to make

sure that any C2-solution of Eq (2.8) stays away from zero; and in view of Lemma2.4,such a conclusion is guaranteed provided infM a > 0.

2.3 A necessary condition for f The purpose of this subsection is to derive a necessary condition for the function f so that (2.8) admits a positive smooth solution

Proposition 2.6 The necessary condition for (2.8) to have positive smooth solutions is that

M f dvg < 0 In particular, the necessary condition for (1.8) to have a positive smooth solution is that

M f dvg < 0.

Proof We assume that u > 0 is a solution of (2.8) By multiplying both sides of (2.8)

by u1−q , and integrating over M, one gets

2.4 The non-existence of smooth positive solutions of suitable small energy Inspired

by [10, Section2] and [17, Subsection 2.5], this subsection is devoted to proving somenon-existence results for smooth positive solution of (1.8) with finite energy In order to

state the result, let us assume that u is a smooth positive solution of (1.8) andα ∈ (0, 1),

β  0 are constants The restriction on α is given as follows

By integrating both sides of (1.8), one obtains

Thus, we have proved that

Second, ifα = 2  /(22 + 1), we recall from (2.13) withβ = 0 that the inequality

then by collecting (2.14) and (2.16), we have proved the following

Proposition 2.7 Let (M, g) be a smooth compact Riemannian manifold of dimension

solutions of (1.8) must have u H1  .

In view of proposition2.7, it is reasonable and necessary to have some control onthe integral

M a dvgas we did in Theorem1.1 However, it is not so clear if a conditionindependent of any norm of solutions could be available as in the positive case, see [10,Theorem 2.1] We note that even in the positive case, the non-existence result was not

able to cover the case when f is non-positive somewhere.

3 The Analysis of the Energy Functionals When supM f > 0

As indicated in the title of this section, we mainly consider the energy functional ciated to (2.8) in the case supM f > 0 As such, unless otherwise stated, we always

asso-assume supM f > 0.

3.1 Functional setting For each q ∈ (2, 2  ) and k > 0, we introduce B k ,q a

hyper-surface of H1(M) which is defined by

B k ,q = ...

the original idea of our approach was based on a mini-max method in a paper by Rauzy[21] However, we find that in the case considered in [21], the assumption of the negativeYamabe invariant... 0

As indicated in the title of this section, we mainly consider the energy functional ciated to (2.8) in the case supM f > As such, unless otherwise stated, we always... necessary condition for f The purpose of this subsection is to derive a necessary condition for the function f so that (2.8) admits a positive smooth solution

Proposition 2.6 The necessary