Separable infinite harmonic functions in cones

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Marie-Françoise Bidaut-Véron, Marta Garcia-Huidobro, Laurent Véron

To cite this version:

Marie-Françoise Bidaut-Véron, Marta Garcia-Huidobro, Laurent Véron. Separable infinite harmonic functions in cones. Calculus of Variations and Partial Differential Equations, Springer Verlag, In press, 35, pp.35 - 62. �hal-01492362v3�

Marie-Françoise Bidaut-Véron Marta Garcia-Huidobro

Laurent Véron

AbstractWe study the existence of separable infinity harmonic functions in any cone ofR^N vanishing on its boundary under the formu(r, σ) =r^−βψ(σ). We prove that such solutions exist, the spherical partψsatisfies a nonlinear eigenvalue problem on a subdomain of the sphereS^N−1 and that the exponentsβ = β+ > 0 and β = β₋ < 0 are uniquely determined if the domain is smooth. We extend some of our results to non-smooth domains.

2010 Mathematics Subject Classification.35D40; 35J70; 35J62.

Key words.Infinity-Laplacian operator; large solutions; viscosity solutions; comparison; ergodic constant; boundary Harnack inequality.

Contents

1 Introduction 1

2 The smooth case 4

2.1 Two-sided estimates . . . 5

2.2 Gradient estimates . . . 9

2.3 Proof of Theorem A . . . 12

2.4 Proof of Theorem B . . . 14

3 The general case 16 3.1 Problem on the circle . . . 16

3.2 The spherical cap problem . . . 18

3.3 Proof of Theorem D . . . 22

3.4 Proof of Theorems E and F . . . 28

1 Introduction

LetSbe aC³ subdomain of the unit sphereS^N−1 ofR^N andC_S :={λσ∈R^N :λ >0, σ∈S}is the positive cone generated byS. In this paper we study the existence of positive solutions of

∆∞u:= 1

2∇ |∇u|².∇u= 0 (1.1)

1

inC_Svanishing on∂C_S\ {0}under the form

u(x) =u(r, σ) =r^−βψ(σ), (1.2)

whereβ ∈Rand(r, σ) ∈ R+×S^N−1 are the spherical coordinatesR^N; such a functionuis called a separable infinity harmonic function. The functionψsatisfies thespherical infinity harmonic problem in

S 1

2∇⁰|∇⁰ψ|².∇⁰ψ+β(2β+ 1)|∇⁰ψ|²ψ+β³(β+ 1)ψ³ = 0 inS ψ= 0 on∂S,

(1.3) where ∇⁰ is the covariant gradient onS^N−1 for the canonical metric and(a, b) 7→ a.b the associated quadratic form. The role of the infinity Laplacian for Lipschitz extension of Lipchitz continuous func- tions defined in a domain has been pointed out by Aronsson in his seminal paper [1]. When the infinity Laplacian ∆∞ is replaced by the p-Laplacian, the research of regular (β < 0) separable p-harmonic functions has been carried out by Krol [8] in the2-dim case and by Tolksdorff [16] in the general case.

Following Krol’ s method, Kichenassamy and Véron [10] studied the2-dim singular case (β > 0). Fi- nally, by a completely different approach and in a more general setting Porretta and Véron [15] studied the general case. In that case, the functionψsatisfies thesphericalp-harmonic problem inS

div⁰

(β²ψ²+|∇⁰ψ|²)^p−22 ∇⁰ψ

+βλβ(β²ψ²+|∇⁰ψ|²)^p−22 ψ= 0 inS

ψ= 0 on∂S, (1.4)

whereλβ =β(p−1) +p−N anddiv⁰is the divergence operator acting on vector fields inT S^N−1. Following an idea which was introduced by Lasry and Lions [13], Porretta-Véron’s method was to transform the equation(1.4)by setting

w=−1

βlnψ (1.5)

in the caseβ >0. The functionwsatisfies the new problem

−div⁰

1 +|∇⁰w|²p/2−1

∇⁰w

+ 1 +|∇⁰w|²p/2−1

β(p−1)|∇⁰w|²+λ_β

= 0 inS

ρ(σ)→0lim w(σ) =∞, (1.6) whereρ(σ) :=dist(σ, ∂S)is the distance understood in the the geodesic sense onS.

In this article we borrow ideas used in [15] to transform problem(1.1)by introducing the function wdefined by(1.5). Thenwsatisfies, in the viscosity sense,

−1 2∇⁰

∇⁰w

2.∇⁰w+β

∇⁰w

4+ (2β+ 1)

∇⁰w

2+β+ 1 = 0 inS

ρ(σ)→0lim w(σ) =∞. (1.7) We first prove

Theorem A.LetS ⊂S^N−1be a proper subdomain ofS^N−1with aC³ boundary. Then for anyβ >0 there exists a locally Lipschitz continuous functionwand a uniqueλ(β) >0satisfying in the viscosity sense

−1 2∇⁰

∇⁰w

2.∇⁰w+β

∇⁰w

4+ (2β+ 1)

∇⁰w

2+λ(β) = 0 inS

ρ(σ)→0lim w(σ) =∞. (1.8) whereρ(.)is the geodesic distance from points inSto∂S.

Then we prove that there exists a uniqueβ such thatλ(β) =β+ 1. In a similar way we study the regular case whereβ <0in(1.2), (we denote−β=µ >0), and we obtain

Theorem B.LetS ⊂S^N−1be subdomain with aC³boundary. Then there exist exactly two real positive numbersβ_sandµ_sand at least two positive functionsψ_sandω_sinC(S)∩C_loc^0,1(S)(up to multiplication by constants) such that the two functionsu_s,+ andu_s,−defined inC_S byu_s,+(r, σ) := r^−βsψ_s(σ)and us,−(r, σ) := r^µsωs(σ) are infinity harmonic in CS and vanish on ∂CS \ {0} and∂CS respectively.

Furthermoreβ_sandµ_sare decreasing functions ofSfor the inclusion order relation on sets.

The previous results can be extended to general regular domains on a Riemannian manifold as in [15]. It is an open problem whether the positive solutions associated to the same exponentβ_s(orb_s) are proportional, see discussion in Remark p. 15.

In the special case of a rotationally symmetric domainSwe have a more precise result which allows us to characterize all the separable infinity harmonic functions in CS which keep a constant sign and vanish on∂C_S\ {0}. We denote byφ∈(0, π)the azimuthal angle from the North poleN onS^N−1. Theorem C.LetSαbe the spherical cap with azimuthal openingα∈(0, π]. Then there exist two positive functionsψ_αandω_αinC^∞(S), vanishing on∂S, such that the two functions

u_α,+(r, σ) =r^{− π}

2

4α(π+α)ψ_α(σ), (1.9)

and

uα,−(r, σ) =r ^π

2

4α(π−α)ω_α(σ), (1.10)

are infinity harmonic inCSαand vanish on∂CSα\ {0}. The two functionsψαandωαdepend only on the variableφ ∈(0, α]and are unique in the class of rotanionnaly invariant solutions up to multiplication by constants.

This study reduced to an ordinary differential equation which has been already treated by T. Bhat- tacharya in [4] and [5]. But for the sake of completeness and for some related problems we present it in Section 3 of the present paper.

Using these previous results we prove the existence of separable infinity harmonic functions in any coneC_S.

Theorem D.AssumeS (S^N−1is any domain. Then there existβ_s >0and a positive functionψ_sin C(S), locally Lipschitz continuous inSand vanishing on∂S, such that the function

us,+(r, σ) =r^−βsψ_s(σ), (1.11) is infinity harmonic inCSand vanishes on∂CS\ {0}.

When the coneC_S is a little more regular, the construction of the spherical infinity harmonic func- tions can be performed via an approximation from outside.

Theorem E.Assume thatS (S^N−1is an outward accessible domain, i.e. ∂S=∂S^c. Then there exist β_s ∈(0, β_s]and a positive functionψ_s∈C(S), locally Lipschitz continuous inSand vanishing on∂S, such that the function

u_s,+(r, σ) =r^−βsψ_s(σ) (1.12) is infinity harmonic inC_S, vanishes on∂C_S\ {0}and has the property that for any separable infinity- harmonic function inC_S under the formu(r, σ) = r^−βψ(σ)whereβ >0andψinC(S)vanishes on

∂S, there holdsβ_s≤β ≤β_s.

The uniqueness of the exponentβsis proved under Lipschitz and geometric conditions onS.

Theorem F.Assume thatS (S^N−1 is a Lipschitz domain satifying the interior sphere condition. Then β_s = β_s. Furthermore there exists a constantc= c(S, p)> 0such that for any two positive functions ψ_i,i= 1,2, satisfying the spherical p-harmonic problem(1.4), there holds

ψ1(σ)

ψ₂(σ) ≤cψ1(σ⁰)

ψ₂(σ⁰) ∀(σ, σ⁰)∈S. (1.13) Note that the statements and the proofs of Theorems D, E and F can be easily modified if one con- siders regular infinity harmonic functions inCSwhich vanish on∂CS.

Acknowledgements. This article has been prepared with the support of the collaboration programs ECOS C14E08 and FONDECYT grant 1160540 for the three authors. The authors are grateful to the referee for a careful reading of their work.

2 The smooth case

We assume that(r, σ)∈R+×S^N−1are the spherical coordinates ofx∈R^N. Ifuis aC¹function, then

∇u=u_re+ ¹_r∇⁰uwheree= _|x|^x and∇⁰is the tangential gradient ofu(r, .)identified to the covariant gradient thanks to the canonical imbedding ofS^N−1intoR^N. Then|∇u|² =u²_r+_r¹2|∇⁰u|², thus

−∆_∞u=−

u²_r+ 1 r²|∇⁰u|²

r

ur− 1 r²∇⁰

u²_r+ 1

r²|∇⁰u|²

.∇⁰u= 0.

A solution−∆_∞u = 0which has the formu(x) =u(r, σ) =r^−βψ(σ)satisfies, in the viscosity sense, the spherical infinity harmonic equation

1

2∇⁰|∇⁰ψ|².∇⁰ψ+β(2β+ 1)|∇⁰ψ|²ψ+β³(β+ 1)ψ³ = 0. (2.1) Theorem 2.1. For anyC³domainS⊂S^N−1there exists a uniqueβs>0and one nonnegative function ψ∈C^0,1(S)solution of

−1

2∇⁰|∇⁰ψ|².∇⁰ψ=β(2β+ 1)|∇⁰ψ|²w+β³(β+ 1)ψ³ in S

ψ= 0 in ∂S.

(2.2)

such that the function(r, σ) 7→ u_s(r, σ) := r^−βsψ(σ)is positive and ∞-harmonic in the coneC_s = {x=λσ∈R^N :λ >0, σ ∈S}and vanish on∂S\ {0}.

Following Porretta-Veron’s method, we transform the eigenvalue problem into a large solution prob- lem with absorption by setting

w=−1

β lnψ. (2.3)

Therefore the formal new problem is to prove the existence of a unique β > 0 and of a nonnegative functionwsuch that

1

2∇⁰|∇⁰w|².∇⁰w−β|∇⁰w|⁴−(2β+ 1)|∇⁰w|²=β+ 1 in S w=∞ in ∂S.

(2.4)

The two problems are clearly equivalent for C² solutions. Since the mapping w 7→ ψ is smooth and decreasing, it exchanges supersolutions (resp. subsolutions) into subsolutions (resp. supersolutions).

Therefore the two problems(2.2)-(2.4)are also equivalent if we deal with continuous viscosity solutions.

In order to increase the regularity of the solutions and to avoid the difficulties coming form the fact the above problem is invariant if we add a constant to a solution, instead of(2.4)we consider the regularized problem with absorption

−δ∆w−1

2∇⁰|∇⁰w|².∇⁰w+γ|∇⁰w|⁴+ (2γ+ 1)|∇⁰w|²+w= 0 in S w=∞ in ∂S,

(2.5) where, δ are two positive parameters. We will obtain below local estimates on∇⁰windependent of andδ. Thanks to these estimates we will let successivelyδ andto0and obtain that, up to a constant, the termw converges to some unique λ(γ) calledthe ergodic constantalthough it has a probabilistic interpretation only in the case of the ordinary Laplacian [13]. The limit problem of(2.5)is the following

−1

2∇⁰|∇⁰w|².∇⁰w+γ|∇⁰w|⁴+ (2γ+ 1)|∇⁰w|²+λ(γ) = 0 in S w=∞ in ∂S.

(2.6)

2.1 Two-sided estimates

We denote the "positive" geodesic distanceρ(σ) =dist(σ, ∂S). Ifσ ∈S^cwe setρ(σ) =˜ −dist(σ, ∂S).

Ifσ1andσ2are not antipodal points there exists a unique minimizing geodesic betweenσ1 andσ2. It is an arc of a Riemannian circle (or great circle). The geodesic distance betweenσ₁ andσ₂is denoted by

`(σ₁, σ₂). It coincides with the angle determined by the two straight lines from0toσ₁ and0toσ₂. At this point it is convenient to use Fermi coordinates inSin a neighborhood of∂S. We set

S_τ ={σ∈S:ρ(σ)< τ}, S_τ⁰ =S\S_τ, Σ_τ ={σ∈S:ρ(σ) =τ}.

If τ ≤ τ₀ for any σ ∈ S_τ there exists a unique z_σ ∈ ∂S such that `(σ, z_σ) = ρ(σ). These Fermi coordinates ofσare defined by(τ, z)∈[0, τ0)×∂S. The mappingΠsuch that

Π(σ) = (ρ(σ), z_σ) ∀σ∈S_τ0,

is aC²diffeomorphism fromS_τ0 into[0, τ₀)×∂S. The expression of the Laplace-Beltrami operator in S_τ0 is given in [3]:

∆⁰u(σ) = ∂²u

∂τ² −(N −2)H∂u

∂τ + ˜∆⁰_zu ∀σ = Π⁻¹((τ, z)), (2.7) whereH = H(τ, z) is the mean curvature ofΣ_τ and∆˜⁰_z is a second order elliptic operator acting on functions defined onΣτ. Ifg = (gij)is the metric tensor onS^N−1 and by convention,|g| =det(gij), this operator admits the following expression

∆˜⁰_zu= 1 p|g|

N−2

X

j=1

∂

∂z_j

p|g|a_j ∂u

∂z_j

,

for somea_j > 0if we take for coordinates curve-framez_j a system of orthogonal 1-dim great circles onΓintersecting atzσ (these circle corresponds to the (N −2)-principal curvatures at this points). The coefficientsa_j depend both onzandτ. Thus, ifudepends only onρ,

∆⁰u(σ) = ∂²u

∂τ² −(N−2)H∂u

∂τ. (2.8)

The expression of H is given in [3] and we can assume that τ0 is small enough so that H remains bounded.

We extend the geodesic distanceρ(x) =dist(x, ∂S)as a smooth positive function so thatρ(x) :=˜ ρ(x) ifρ(x) ≤ τ0 and thus, it the same neighborhood of∂S, ∇˜ρ(x) = nzx, the unit outward normal vector to∂Sat the pointz_x=P roj_∂Ω(x).

Ifwdepends only onρ,(2.5)becomes

δw⁰⁰−(N −2)Hw⁰+w⁰²w⁰⁰−γw⁰⁴−(2γ+ 1)w⁰²−w= 0 in S_τ0

w=∞ in ∂S. (2.9)

In the sequel we put P_δ(u) :=−δ∆u− 1

2∇⁰|∇⁰u|².∇⁰u+γ|∇⁰u|⁴+ (2γ+ 1)|∇⁰u|²+u= ˜P_δ(u) +u. (2.10) Proposition 2.2. There existτ₁ ∈ (0, τ₀], three positive constantsM,₀andδ₀ and two positive func- tionsw^∗, w∗ ∈C²(S)such thatw^∗ > w∗inS_τ,w^∗+1

γ lnρ∈L^∞(S)andw∗+ 1

γ lnρ ∈L^∞(S)with the property that for any∈(0, 0]andδ∈(0, δ0]the two functions

¯

w(σ) =w^∗+ M

(2.11)

and

w(σ) =w∗−M

(2.12)

are respectively a supersolution and a subsolution ofP_δ(u) = 0. Furthermore any solutionwof problem (2.6)satisfiesw≤w≤w.¯

Proof. Leta >0. We first notice by a standard computation that the solutions of the ODE, δw⁰⁰+w⁰²w⁰⁰−γw⁰⁴−aw⁰²= 0 in (0,1)

w⁰(0) =−∞ (2.13)

are negative and given implicitely by δ

aw⁰(ρ) + 1

γ − δ a

rγ atan⁻¹

ra γ

1 w⁰(ρ)

=−ρ. (2.14)

In order to have a global estimate, we setw⁰=−z⁻¹, thus(2.14)becomes δz

a + 1

γ − δ a

rγ atan⁻¹

z

ra γ

=ρ, (2.15)

provideda > γδ. Sincetan⁻¹ zq

a γ

≤zq

a

γ, we derive z(ρ)≥γρ⇐⇒0> w⁰(ρ)≥ − 1

γρ ∀ρ >0, (2.16) with equality only ifρ= 0. Since we can write(2.15)as

rγ a δ a

z

ra γ

+

1 γ − δ

a rγ

atan⁻¹

z ra

γ

=ρ≥tan⁻¹

z ra

γ

, (2.17)

we obtain

z≤ rγ

atan ρ√ aγ

⇐⇒w⁰(ρ)≤ − ra

γ cot ρ√ aγ

∀ρ >0. (2.18) Finally

− 1

γρ ≤w⁰(ρ)≤ − ra

γ cot ρ√ aγ

∀ρ∈(0, τ₀]. (2.19) and in particular, for anyτ1∈(0, τ0],

|w⁰(ρ)| ≥ ra

γ cot τ1

√aγ

∀ρ∈(0, τ1]. (2.20)

From this estimate we derive w(ρ0) + 1

γ ln

sinρ0

√aγ sinρ√

aγ

≤w(ρ)≤w(ρ0) + 1 γln

ρ0

ρ

∀ρ∈(0, τ1]. (2.21)

The solutionwdepends on the value ofaandδ. Since∂Sis smooth, we can assume that(N−2)|H|

is bounded by some constantm ≥ 0inSτ0. Denote bywτ a solution satisfyingw(τ) = 0, then it is positive inSτ and

P_δ(w_τ) =δ(N −2)Hw_τ⁰ + (2γ+ 1−a)w⁰²_τ +w_τ

≥ |w⁰_τ|((2γ+ 1−a)|w⁰_τ| −m)

≥ |w⁰_τ|((2γ+ 1−a) ra

γ cot τ√ aγ

−m).

If we take1< a₁ <2γ+ 1, we chooseτ :=τ₁ ∈(0, τ₀]such that (2γ+ 1−a₁)

ra

γ cot τ√ a₁γ

> m, (2.22)

which implies thatwτ is a supersolution inSτ. In assuming nowa:=a2 >2γ+ 1, we also have, P_δ(w_τ)≤w_τ⁰((2γ+ 1−a)w_τ⁰ −m) +w_τ

≤ |w⁰_τ|((2γ+ 1−a)|w⁰_τ|+m) +w_τ

≤ |w⁰_τ|

m−(a−2γ−1) ra

γ cot τ√ aγ

+w_τ. We choosea₂ = 4γ+ 2−a₁, then

m−(a−2γ−1) ra

γcot (τ√

aγ)≤ −c <0 ∀τ ∈(0, τ1].

Therefore

P_δ(w_τ)≤w_τ

+ w_τ⁰ w_τ

.

Sincew_τ⁰₁ <0andw_τ1(τ₁) = 0⁺, there exists₀>0, such that for any∈(0, ₀] +w⁰_τ1(ρ)

wτ1(ρ) ≤ −1 ∀ρ∈(0, τ₁].

Thereforewτ1,a1 andwτ1,a2 are respectively supersolution and subsolution ofP_δ(u) = 0 inSτ1. We extend them inS_τ⁰₁ as smooth functionsw˜τ1,a1 andw˜τ1,a2 in order

P˜_δ( ˜wτ1,aj)

to remain bounded by some constantM. Finallyw¯ = ˜wτ1,a1+M ⁻¹is a supersolution andw= ˜wτ1,a2+M ⁻¹is a subsolution ofP_δ(u) = 0.

Next, we replacewby

w_h(δ) =w(δ+h) andw¯by

¯

w_h = ¯w(δ−h)

forh small enough, we still have a sub and a super solution ofP_δ(u) = 0 inSτ1 andSτ1 \S_h. In the remaining part of S, we extend smoothly w_h andw¯_h in order P˜_δ(w_h) andP˜_δ( ¯w_h) be bounded. We can adjust M in orderP_δ(w_h) ≤ 0andP_δ( ¯wh) ≥ 0 in wholeS, and all these manipulations can be

done uniformly with respect tohand. Ifwis anyC² solution of(2.5), we prove that it dominates the subsolutionw_hinS: actually, if we assume thatw_handware not ordered inS, there existsσ0∈Ssuch that

w_h(σ₀)−w(σ₀) = max{w_h(σ)−w(σ) :σ∈S}>0.

Since the two functions areC²,

∇w_h(σ₀) =∇w(σ₀) and D²w_h(σ₀)≤D²w(σ₀), whereD² is the Hessian form, in the sense of quadratic forms, i.e.

D²w_h(σ0)(∇w_h(σ0),∇w_h(σ0))≤D²w(σ0)(∇w(σ₀),∇w(σ₀)).

This impliesP_δ(w_h)(σ₀)>P_δ(w)(σ₀) = 0, contradiction. Therefore

w_h ≤w in S, (2.23)

uniformly with respect toh. Similarly

¯

wh≥w in S. (2.24)

Lettinghtend to0the claim follows.

2.2 Gradient estimates

Ifσ0∈S^N−1andR < π, we setBR(σ0) ={σ∈S^N−1:`(σ, σ0)< R}.

Proposition 2.3. Let0≤δ, ≤1andwbe a smooth solution of

−1

2∇⁰|∇⁰w|².∇w−δ∆w+γ|∇⁰w|⁴+ (2γ+ 1)|∇⁰w|²+w = 0 in B_R(σ₀)⊂S, (2.25) whereSis a domain ofS^N−1. Then there existsc=c(N)>0such that

|∇⁰w(σ0)| ≤ c

γR. (2.26)

Proof. We setz=|∇w|², then2∆∞w=∇⁰|∇⁰w|².∇⁰w=∇⁰z.∇⁰w. We define the linearized operator of∆∞atwfollowinghby

Bw(h) := d

dt∆∞(w+th)b_t=0= 1

2∇⁰h.∇⁰z+∇⁰w.∇⁰(∇⁰w.∇⁰h).

Thus the linearized operator of∆∞+δ∆atwfollowinghis

L_w(h) =B_w(h) +δ∆h. (2.27)

Thus

L_w(z) = 1

2|∇⁰z|²+∇⁰w.∇⁰(∇⁰.∇z) +δ∆z.

We can re-write(2.25)under the form

∇⁰w.∇z= 2 γz²+ (2γ+ 1)z+w−δ∆w

. (2.28)

Hence

∇⁰(∇⁰w.∇⁰z) = 2 (2γz+ 2γ+ 1)∇⁰z+∇⁰w−δ∇⁰(∆w) , and then

∇w.∇⁰(∇⁰w.∇⁰z) = 2 (2γz+ 2γ+ 1)∇⁰z.∇⁰w+z−δ∇⁰(∆w).∇⁰w .

By the Weitznböck formula, sinceRicc (S^N−1) = (N−2)g0(g0is the metric tensor onS^N−1), we have 1

2∆z= D²w

2+∇⁰(∆w).∇⁰w+ (N−2)|∇⁰w|²

= D²w

2+∇⁰(∆w).∇⁰w+ (N−2)z.

Hence

L_w(z) = 1

2|∇z|²+ 2 ((2γz+ 2γ+ 1)∇⁰z.∇⁰w+ 2z−2δ∇⁰w.∇⁰(∆w)) + 2δ

D²w

2+ 2δ∇⁰(∆w).∇⁰w+ 2δ(N−2)z.

Expanding the above identity, we see that the terms of order 3 disappear, hence L_w(z) = 1

2|∇z|²+ 2 ((2γz+ 2γ+ 1)∇⁰z.∇⁰w+ 2z) + 2δ D²w

2+ 2δ(N −2)z. (2.29) Ifξ ∈C_c²(BR(σ0)), we setZ =ξ²zand we derive

L_w(Z) =Bw(ξ²z) +δ∆(ξ²z)

=ξ²L_w(z) +zL_wξ²+ 2(∇⁰w.∇⁰ξ²)(∇⁰w.∇z) + 2δ∇⁰ξ².∇z, where

L_wξ² = 1

2∇⁰z.∇⁰ξ²+∇⁰w.∇⁰(∇⁰w.∇⁰ξ²) +δ∆ξ²

= 1

2∇⁰z.∇⁰ξ²+D²w(∇⁰ξ²).∇⁰w+D²ξ²(∇⁰w).∇⁰w+δ∆ξ²

=∇⁰z.∇⁰ξ²+D²ξ²(∇⁰w).∇⁰w+δ∆ξ²

≥ ∇⁰z.∇⁰ξ²−z D²ξ²

+δ∆ξ². By Schwarz inequality,(∆w)²≤ _N−1¹

D²w

2, we derive from(2.27)and(2.28), L_w(z)≥ 1

2|∇z|²+ 4 (2γz+ 2γ+ 1) γz²+ (2γ+ 1)z+w−δ∆w + 4z + 2δ

N−1(∆w)²+ 2δ(N−2)z

≥ 1

2|∇z|²+ δ

N−1(∆w)²+ 4γ²z³−c₀,

for somec₀ =c₀(N, γ) >0. In the sequel the different positive constantsc_j which will appear bellow depend only onN andγ. This implies

L_w(Z)≥z ∇⁰z.∇⁰ξ²−z D²ξ²

+δ∆ξ²

+ 2(∇⁰w.∇⁰ξ²)(∇⁰w.∇z) + 2δ∇⁰ξ².∇z +ξ²

1

2|∇z|²+ δ

N−1(∆w)²+ 4γ²z³−c₀

. (2.30)

We chooseξsuch that0≤ξ≤1,|∇ξ| ≤c1R⁻¹and D²ξ

≤c1R⁻², then (z+ 2δ)

∇⁰z.∇⁰ξ² ≤c1

(z+ 2δ)ξ R

∇⁰z ≤ ξ²

8 ∇⁰z

2+c2

(z+ 2δ)² R² , z δ∆ξ²−z

D²ξ²

≤ c3(z+ 2δ)² R² , ∇⁰w.∇z

≤√ z|∇z|, ∇⁰w.∇ξ²

≤2ξ|∇w| |∇ξ| ≤ 2c1ξ√ z R , 2(∇⁰w.∇⁰ξ²)(∇⁰w.∇z)

≤ 4c₁ξz|∇z|

R ≤ ξ² 8

∇⁰z

2+c₄z² R².

We consider a pointz₀ ∈ B_R where Z is maximal, thenL_w(Z)(z₀) ≤ 0, which implies that at this point,

ξ² 1

2|∇z|²+ δ

N −1(∆w)²+ 4γ²z³−c₀

≤ ξ²

4 |∇z|²+c₅(z+ 2δ)²

R² . (2.31)

We assumeR≤1and2δ≤1, we multiply byξ⁴and obtain 1

4

ξ³∇z

2+ δξ⁶

N−1(∆w)²+ 4γ²(ξ²z)³ ≤ c₆((ξ²z)²+ 1)

R² +c₀. (2.32)

From the inequality

4γ²(ξ²z)³ ≤ c₆(ξ²z)² R² + c₇

R², we deduce

ξ²z≤ c8

R² withc8 = max n

c7,^c_γ⁶2

o

. (2.33)

If we assume thatξ(σ0) = 1, we finally infer

|∇w(σ₀)| ≤

√c₈

R , (2.34)

which is the claim.

As an immediate consequence, we have

Corollary 2.4. Let0≤, δ ≤1. Ifwis a solution of(2.5)inS, it satisfies

|∇w(σ)| ≤ c

γρ(σ) ∀σ∈S, (2.35)

for somec >0depending only ofN.

2.3 Proof of Theorem A We writew=wδ,,γ and

−δ∆w−1

2∇ |∇⁰w|²∇⁰w+γ|∇⁰w|⁴+ (2γ+ 1)|∇⁰w|²+w = 0, (2.36) thenwδ,,γ satisfies the estimate(2.35). By Proposition 2.2 it satisfies also

−1

γ lnρ−M

≤w∗≤w_δ,,γ ≤w^∗+M ≤ −1

γlnρ+M

. (2.37)

The set of functions {w_δ,,γ}_,δ is clearly locally equicontinuous inS. By classical stability results on viscosity solutions (see e.g. [9, Chap 3]), there exist a subsequence{w_δn,,γ}and a functionw,γ such thatw_δn,,γ →w_,γ, andw_,γis a viscosity solution of

−1

2∇⁰|∇⁰w|².∇⁰w+β|∇⁰w|⁴+ (2β+ 1)|∇⁰w|²+w= 0 in S w=∞ in ∂S.

(2.38)

Furthermore w,γ satisfies the same estimates (2.35)and (2.37)as w_δ,,γ. Putw˜,γ(σ) = w,γ(σ)− w_,γ(σ₀)withσ₀ ∈Ω, thenw˜ := ˜w_,γ satisfies

−1

2∇ |∇⁰w|˜²∇⁰w˜+γ|∇⁰w|˜⁴+ (2γ+ 1)|∇⁰w|˜²+w˜+w(σ0) = 0. (2.39) Moreover

|w˜_,γ(σ)|=|w_,γ(σ)−w_,γ(σ₀)| ≤max c

γρ(τ) :τ ∈[σ, σ₀]

|σ−σ₀|.

Thus, as→ 0,w˜,γ → 0locally uniformly inS. Up to some subsequence{_n},w˜n,γ → wγlocally uniformly in S and_nw_n,γ(σ₀) → λ(γ). As in [15] the expressionλ(γ) does not depend onσ₀. By analogy with the semilinear case studied in [13], this last limit is calledthe ergodic constant. Furthermore it is easy to check that there exist positive constantsM1andM2such thatw1 andw2defined by

w₁(x) =−1

γ lnρ+M₁ρ+M₂ and w₂(x) =−1

γ lnρ−M₁ρ−M₂ (2.40) are respectively a supersolution and subsolution of(2.39)inSand that there holds

w₂(x)≤w˜_,γ(x)≤w₁(x) ∀x∈S. (2.41)

By the same stability results of viscosity solutions, we infer thatw_γis a positive solution of

−1

2∇ |∇⁰w|²∇⁰w+γ|∇⁰w|⁴+ (2γ+ 1)|∇⁰w|²+λ(γ) = 0 inS w=∞ on∂S.

(2.42)

Furthermore, there holds from(2.41)and(2.35),

w_γ+ 1 γ lnρ

≤M, (2.43)

and

|∇w_γ| ≤ c

γρ. (2.44)

Proposition 2.5. For anyC³ domain S ⊂ S^N−1, the ergodic constant λ(γ) := λ(γ, S) is uniquely determined byγ. Furthermore it is a continuous decreasing function ofγandSfor the order relation of inclusion.

Proof. Assume that the set{w(σ0)}of values of the solutions of(2.42)atσ0admits two different clus- ter pointsλ1andλ2. Then there exist two locally Liptchitz continuous functionsw1andw2satisfying

−1

2∇⁰|∇⁰w_i|².∇⁰w_i+γ|∇⁰w_i|⁴+ (2γ+ 1)|∇⁰w_i|²+λ_i = 0 inS, (2.45) in the viscosity sense, and such that

w_i(σ) =−1

γ lnρ(σ) (1 +o(1)) asρ(σ)→0. (2.46) We can assume thatλ₁ > λ₂. For >0letv = (1 +)w₂. Then

−1

2∇⁰|∇⁰v|².∇⁰v+ (1 +)⁻¹γ|∇⁰v|⁴+ (1 +)(2γ+ 1)|∇⁰v|²+ (1 +)³λ₂ = 0 inS. (2.47) ForX >0, we put

f(X) = γ

1 +X²−(2γ + 1)X +λ₁−(1 +)³λ₂. Then

f(X)≥f(X₀) =f

(2γ+ 1)(1 +) 2γ

=−(1 +)(2γ+ 1)²

4γ +λ₁−(1 +)³λ₂. Therefore there exists₀ >0such that for anyX≥0,f(X)≥0, or equivalently

(1 +)⁻¹γX²+ (1 +)(2γ+ 1)X+ (1 +)³λ₂≤γX²+ (2γ+ 1)X+λ₁. (2.48) This implies that

−1

2∇⁰|∇⁰v|².∇⁰v+γ|∇⁰v|⁴+ (2γ+ 1)|∇⁰v|²+λ1 ≥0 inS, (2.49) in the viscosity sense. Sincew1 < vnear∂S, it follows from comparison principle thatw1 < vinS.

Letting→0yields

w₁≤w₂ inS. (2.50)

Since for anyk∈R,w₁+ksatisfies the same equation asw₁and the same estimate(2.46)as thew_iwe obtain a contradiction. Thusλ=λ(γ)is uniquely determined.

For proving monotonicity, assumeγ₁ > γ₂>0and letw_,1andw_,2be solutions of

−1

2∇⁰|∇⁰w,i|².∇⁰w,i+γi|∇⁰w,i|⁴+ (2γi+ 1)|∇⁰w,i|²+w,i = 0 inS, (2.51) such that

w_,i(σ) =−1

γ_i lnρ(σ) (1 +o(1)) asρ(σ)→0. (2.52) Then

−1

2∇⁰|∇⁰w_,1|².∇⁰w_,1+γ₂|∇⁰w_,1|⁴+ (2γ₂+ 1)|∇⁰w_,1|²+w_,1≤0.

Since w,1 ≤ w,2 near∂S, it follows by comparison principle that w,1 ≤ w,2 inS and in particular w_,1 ≤w_,2. Sinceλ₁ = limn→∞w_n,1(x₀)andλ₂ = limn→∞w_n,2(x₀), we infer thatλ₁≤λ₂.

For proving the continuity, let {γ_n} be a sequence converging to γ and let w_n be corresponding solutions of

−1

2∇⁰|∇⁰wn|².∇⁰wn+γn|∇⁰wn|⁴+ (2γn+ 1)|∇⁰wn|²+λ(γn) = 0 inS, (2.53) subject to

wn(σ) + 1

γ_nlnρ(σ)

≤K, (2.54)

for someK >0independent ofn. Since{w_n}is locally bounded inW_loc^1,∞(Ω)we can extract sequences, denoted by{w_nk},{λ(w_nk)}such thatλ(wnk)→λ¯andwnkconverges locally uniformly to a viscosity solutionwof

−1

2∇⁰|∇⁰w|².∇⁰w+γ|∇⁰w|⁴+ (2γ+ 1)|∇⁰w|²+ ¯λ= 0 inS, (2.55) subject tow(σ) = −1

γ lnρ(σ) (1 +o(1)) asρ(σ) → 0. The existence of such a function implies that λ¯=λ(γ). Thus the whole sequence{λ(γ_n)}converges toλ(γ), a fact which implies the continuity.

Next, letS1 ⊂S2be twoC³subdomains ofS^N−1. We denote bywδ,,γ,Sj,j= 1,2, the solutions of (2.5)respectively inS1andS2. Since these solutions are limit of solutions with finite boundary values and that the maximum principle holds, we infer that w_δ,,γ,S2 ≤ w_δ,,γ,S1 inS₁. Lettingδ → 0yields w,γ,S2 ≤ w,γ,S1. Taking σ0 ∈ S1 and since the ergodic constant is uniquely determined, we have w_,γ,S2(σ0)≤w_,γ,S1(σ0)and thusλ(γ, S2)≤λ(γ, S1).

2.4 Proof of Theorem B

We prove below the following proposition using the result of Theorem C, which proof does not depend on the previous constructions.

Proposition 2.6. For anyC³domainS⊂S^N−1, there exists a uniqueβ:=β_ssuch thatλ(β) =β+ 1.

Furthermoreβsis a decreasing function ofSfor the order relation between spherical domains.

Proof. The functionγ 7→λ(γ, S)−γis continuous and decreasing. For >0we consider two spherical capsSi ⊂S ⊂Se; by Proposition 2.5

λ(γ, S_e)≤λ(γ, S)≤λ(γ, S_i), (2.56) then

λ(γ, S_e)−γ ≤λ(γ, S)−γ ≤λ(γ, S_i)−γ. (2.57) By Theorem C, there existsγ =βse andγ=βsisuch that

λ(β_se, S_e)−β_se = 1 and λ(β_si, S_i)−β_si = 1, andβ_se < β_siunlessλ(β_se, S_e) =λ(β_si, S_i)andS_i =S_e. This implies that

λ(βse, S)−βse ≥1 and λ(βsi, S)−βsi ≤1. (2.58) By continuity there exists a uniqueβ =βs ∈[βse, βsi]such thatλ(βs, S)−βs = 1. To this exponent βcorresponds a locally Lipschitz continuous functionwsolution of problem(2.4). Thenψ=e^−βwis a viscosity solution of(2.2). Notice also that the construction ofβ_sand the monotonicity ofS 7→λ(γ, S) imply thatS 7→βsis decreasing.

Similarly we can consider separable infinity harmonic functions under the form(1.2)with negative β <0. We setβ˜=−β, then(2.2)is replaced by

−1

2∇⁰|∇⁰ψ|².∇⁰ψ=µ(2µ−1)|∇⁰ψ|²w+µ³(µ−1)ψ³ in S

ψ= 0 in ∂S.

(2.59)

Ifψis a positive solution of(2.59), we set

w=−1 µlnψ.

Thenwsatisfies

−1 2∇⁰

∇⁰w

2.∇⁰w+µ

∇⁰w

4+ (2µ−1)

∇⁰w

2=µ−1 inS

ρ(σ)→0lim w(σ) =∞, (2.60)

This equation is treated similarly as(2.4).

Remark. It is an open problem whether the positive functions which satisfy(2.2)are unique up to the multiplication by a constant. This is a sharp contrast with the spherical p-harmonic problem with1 <

p < ∞ where uniqueness is proved by the strong maximum principle and Hopf boundary lemma, and this uniqueness result has been extended to Lipschitz domain in [7] using the characterization of the p- Martin boundary obtained by [14], and a sharp version of boundary Harnack principle. See also Section 3.3 for related results. Notice that uniqueness holds when the solutions are spherically radial (see Section 3.2).

3 The general case

3.1 Problem on the circle

We consider here the special caseN = 2andSis the circle in (2.2 ). Fork∈N∗, we set βk= k²

2k+ 1 and µk= k²

2k−1. (3.1)

Proposition 3.1. For anyk ∈ N^∗ there exists two ^π_k-anti-periodicC¹ functionsψ_kandω_kpositive on (0,^π_k)such thatx7→ |x|^−βkψ_k(_|x|^x)is infinity harmonic and singular inR²\ {0}andx7→ |x|^µkω_k(_|x|^x) is infinity harmonic and regular inR².

Proof. We write∇⁰ψ=ψσe^⊥withS¹∼R/2π. Thus(2.2)becomes

−ψ_σ²ψσσ =β³(β+ 1)ψ³+β(2β+ 1)ψ²_σψ, ψ(0) =ψ(2π). (3.2) Forψ6= 0the equation in (3.2 ) can be written as

−ψ_σ² ψ²

ψσσ

ψ =β(2β+ 1)ψ²_σ

ψ² +β³(β+ 1).

We setY = ψ_σ

ψ , thenY_σ+Y² = ψ_σσ ψ and

−Y²Y_σ =Y⁴+β(2β+ 1)Y²+β³(β+ 1) = (Y²+β²)(Y²+β(β+ 1)). (3.3) We first search for solutions withβsuch thatβ(β+ 1)≥0,β 6= 0. Standard computation yields

β

Y²+β² − β+ 1 Y²+β(β+ 1)

Yσ = 1, (3.4)

and this equation is not degenerate and equivalent to(3.2)as long asY 6= 0. Ifβ =−1(that isβ˜₁ in (3.1 )) then(3.4)becomes

Y_σ

Y²+ 1 =−1, (3.5)

thus

tan⁻¹Y(σ) =−σ =⇒Y(σ) =−tanσ=⇒ψ(σ) = sinσ. (3.6) This corresponds to the fact that the coordinate functions are separable and infinity harmonic.

We assume nowβ(β+ 1)>0, or equivalently eitherβ >0orβ <−1. We fixψ(0) = 0and consider an interval on the right of0whereψ > 0. From the equationψ is concave, thusψ_σ(0) >0. Because of concavity and periodicityψmust change sign. We assume thatσ =αis the first critical point ofψ which is a singular point for(3.2)and(3.4). We integrate(3.4)on a small interval(α, σ)and get

tan⁻¹

Y(σ) β

− s

β+ 1

β tan⁻¹ Y(σ) pβ(β+ 1)

!

=σ−α.

Expandingtan⁻¹(x)nearx= 0we obtain Y(σ) =−βp³

3β+ 3√³

σ−α(1 +o(1)) =⇒ψ(σ) =ψ(α)−C(β)ψ(α)(σ−α)⁴³ (1 +o(1)), (3.7) withC(β) = ³

43

4 β√³

β+ 1, and define Y(σ)for σ ∈ (α,2α)by imposing Y(σ) = −Y(2α−σ)and continue this process in order to construct a2α-antiperiodic solution belonging toC^1,13(R). Since

"

tan⁻¹ Y

β

− s

β+ 1

β tan⁻¹ Y pβ(β+ 1)

!#σ=α

σ=0

=α,

withY(0) =∞,Y(α) = 0, the condition forπ-antiperiodicity is therefore s

β+ 1 β −1

! π

2 =α⇐⇒β = π² 4(α²+απ). Ifβ >0, the periodicity condition yields

rβ+ 1

β = 1 + 1

k ⇐⇒β =β_k = k²

1 + 2k. (3.8)

Ifβ <−1

1− s

β+ 1 β

! π=α, and the periodicity condition implies

rβ+ 1

β = 1− 1

k ⇐⇒β = ˜β_k = k²

1−2k. (3.9)

The caseβ(β+ 1) < 0, or equivalently−1 < β < 0, is easily ruled out. We find that (3.3 ) has the constant solutionY(σ) =p

−β(β+ 1), meaningψ(σ) =Cexp(σp

−β(β+ 1)), which is by no mean periodic. On the other hand, in this case we can write(3.4)under the form

d

dσ tan⁻¹ Y

β

−1 2

rβ+ 1

−β ln |Y −p

−β(β+ 1)|

Y +p

−β(β+ 1)

!!

= 1. (3.10)

Since Y runs fromY(0) =∞ toY(α) = 0, there must be a valueσ₀ whereY(σ₀) = p

−β(β+ 1).

We can integrate (3.10 ) on(0, σ0−)and let→0. Sinceβ <0, it yields π

2 + tan⁻¹

Y(σ₀−) β

−1 2

rβ+ 1

−β ln |Y(σ₀−)−p

−β(β+ 1)|

Y(σ0−) +p

−β(β+ 1)

!

=σ₀−.

The left-hand side expression tends to∞when→0, a contradiction. Hence there are no solutions with

β ∈(−1,0). This ends the proof of the proposition.

Remark. Whenk= 1the coordinate functions are infinity harmonic and vanish on a straight line. When k= 2, the regular solution withµ1= ⁴₃ is

u(x, y) =x⁴³ −y⁴³.

Its existence is due to Aronsson [2]. The corresponding circular function,ω(σ) = (cosσ)⁴³ −(sinσ)⁴³, admits four nodal sets on S¹. Whenk = 1, thenβ₁ = ¹₃. It is proved in [4] that any positive infinity harmonic function in a half-space which vanishes on the boundary except at one point blows-up like the separable infinity harmonic functionu(r, σ) =r⁻¹³ψ(σ).

3.2 The spherical cap problem

Proof of Theorem C. The following representation ofS^N−1is classical S^N−1=

σ = (sinφ σ⁰,cosφ) :σ⁰ ∈S^N−2, φ∈[0, π] .

Then∇⁰ψ=ψ_φe+∇⁰_σ0ψwhereeis a tangent unit downward vector toS^N−1following the great circle going through the pointσ. Then|∇⁰ψ|² =ψ_φ²+|∇⁰_σ0ψ|², thus, ifψdepends only onφ, we have

1

2∇⁰|∇⁰ψ|².∇⁰ψ=ψ_φφψ²_φ.

Therefore such a functionψ, if it is aC¹solution of(2.2)in the spherical capS_αdefined forφ∈(0, α), satisfies

−ψ_φφψ_φ² =β(2β+ 1)ψ_φ²ψφ+β³(β+ 1)ψ in(0, α)

ψ_φ(0) = 0, ψ(α) = 0. (3.11)

The conclusion follows from Proposition 3.1.

Remark. Ifα=π, the exponentβ+is ¹₈ andψ:=ψ

Σc is a positive solution of

−ψ_σ²ψσσ = 9

4096ψ³+ 5

32ψ²_σψ in(−π, π) ψ(−π) =ψ(π) = 0.

(3.12)

Then the functionu

Σc(r, σ) =r⁻¹⁸ψ

Σc(σ)is an infinity harmonic function inR^N\L

Σc, which vanishes on the half lineL

Σc :={x =tΣ :t≥0}. The functionY = ^ψ_ψ^σ can be computed implicitly on(0, π) thanks to the identity

tan⁻¹(8Y(σ))−3 tan⁻¹(⁸₃Y(σ)) =σ. (3.13) This yields, withZ = ^8Y₃ ,tan⁻¹(3Z(σ))−3 tan⁻¹(Z(σ)) =σ, hence

3Z−tan(3 tan⁻¹(Z)))

1 + 3Ztan(3 tan⁻¹(Z)) = tanσ, (3.14) since

tan(3x) = 3 tanx−tan³x 1−3 tan²x .

This yields

−8Z³

1 + 6Z²−3Z⁴ = tanσ, (3.15)

which gives the value ofY by solving a fourth degree equation and thenψ=ψ

Σc by integratingY. Using Theorem C we can prove the existence of a singular infinity harmonic function in a coneC_Sκ,α generated by a spherical annulusS_κ,αof the spherical points with azimuthal angleκ < φ < α.

Proposition 3.2. Assume 0 ≤ κ < α < πand letν = ¹₂(α−κ). Then there exists a positive singu- lar infinity harmonic function u_Sκ,α,+ and a regular infinity harmonic functionu_sκ,α,− inC_Sκ,α which vanish respectively on∂C_Sκ,α \ {0}and∂C_Sκ,α under the formu_Sκ,α,+(r, σ) = r^−βsκ,αψ_Sκ,α(σ)and u_Sκ,α,−(r, σ) =r^µSκ,αω_Sκ,α(σ)where

β_Sκ,α = π²

4ν(π+ν) and ω_Sκ,α = π²

4ν(π−ν), (3.16)

andψ_Sκ,α andω_Sκ,α are positive solutions of (3.2)inSκ,αvanishing atκ andα with β = β_Sκ,α and β =−µ_Sκ,α respectively.

Proof. By Theorem C there exists a positive and even solutionψ˜of

ψ˜_φφψ_φ² =β(2β+ 1) ˜ψ_φ²ψ˜_φ+β³(β+ 1) ˜ψ in(−ν, ν) := ¹₂(κ−α),¹₂(α−κ)

ψ(−ν) = 0, ψ(ν) = 0, (3.17)

withβ =β_Sκ,α orβ =−µ_Sκ,α. Thenφ7→ψ(φ) := ˜ψ(φ+ ¹₂(κ+α))is a positive solution of(3.2)in

(κ, α). The proof follows.

The next technical lemma is a variant of Theorem C and Proposition 3.2.

Lemma 3.3. Assume0< α < πand, γ >0. Then the solutionv=v,γ,αof

−v⁰²v⁰⁰+γv⁰⁴+ (2γ+ 1)v⁰²+v= 0 in(0, α)

v(0) =∞, v(α) =∞, (3.18)

is an increasing function of. If 0 < σ0 < α, there existsλ= λ(α, γ) = lim→0v,γ,α(σ0) and this value is independent ofσ₀. The functionv˜= ˜v_,γ,α=v_,γ,α−v_,γ,α(σ₀)converges locally uniformly in (0, α)to a solutionv=v_γ,αof

−v⁰²v⁰⁰+γv⁰⁴+ (2γ + 1)v⁰²+λ= 0 in(0, α)

v(0) =∞, v(α) =∞, (3.19)

with

λ(α, γ) = 1 4γ³

π²

α² −γ(2γ+ 1) 2

. (3.20)

Furthermore

v_γ,α(φ) =−1

γ lnφ as φ→0, (3.21)

and

v_γ,α(φ) =−1

γ ln(α−φ) as φ→α. (3.22)