Quasilinear Elliptic Hamilton-Jacobi Equations on Complete Manifolds

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Quasilinear Elliptic Hamilton-Jacobi Equations on Complete Manifolds

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

To cite this version:

Marie-Françoise Bidaut-Veron, Marta Garcia-Huidobro, Laurent Veron. Quasilinear Elliptic

Hamilton-Jacobi Equations on Complete Manifolds. 2013. �hal-00827016v3�

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QUASILINEAR ELLIPTIC HAMILTON-JACOBI EQUATIONS ON COMPLETE MANIFOLDS

Marie-Fran¸coise Bidaut-V´eron¹, Marta Garcia-Huidobro², Laurent V´eron³

R´esum´e. Let (Mⁿ, g) be an-dimensional complete, non-compact and connected Riemannian manifold, with Ricci tensor Riccg and sectional curvature Secg. AssumeRiccg≥(1−n)B², and eitherp >2 andSecg(x) =o(dist²(x, a)) when dist²(x, a) → ∞ for a ∈ M, or 1 p−1 > 0, any C¹ solution of (E)−∆pu+|∇u|^q = 0 on M satisfies

|∇u(x)| ≤cn,p,qB^q+1−p¹ for some constantcn,p,q>0. As a consequence there existscn,p >0 such that any positive p-harmonic function von M satisfies v(a)e⁻^c^n,p^Bdist(x,a)≤v(x)≤v(a)e^c^n,p^Bdist(x,a) for any (a, x)∈M×M.

Equations de Hamilton-Jacobi quasilinéaires sur une variété complète

Résumé. Soit (Mⁿ, g) une variété riemanniennen-dimensionnelle complète, non compacte et connexe de courbures de RicciRiccget sectionnelleSecg. On supposeRiccg≥(1−n)B²etSecg(x) =o(dist²(x, a)) sidist²(x, a)→ ∞pour a∈Msip >2, ouSecg(x)≤0 si 1 p−1>0, toute solution de classeC¹de (E)−∆pu+|∇u|^q= 0 surM satisfait à|∇u(x)| ≤cn,p,qB^q+1−p¹ où cn,p,q >0 est une constante. On en déduit qu’il existecn,p > 0 tel que toute fonctionp-harmonique positivevsurMsatisfait à l’encadrement suivant, v(a)e⁻^c^n,p^Bdist(x,a)≤v(x)≤v(a)e^c^n,p^Bdist(x,a)pour tout (a, x)∈M×M.

Version fran¸caise abrégée. Soit (Mⁿ, g) une variété riemannienne complète, non-compacte et connexe de courbure de RicciRicc_get courbure sectionnelleSec_g. Pour tout p >1, on dénote paru7→∆pu:=div

|∇u|^p⁻²∇u

lep-Laplacien sur M pour la m´etriqueg. Notre r´esultat principal est le suivant

Theor`eme 1. Soit B≥0 tel queRiccg≥(1−n)B² et q > p−1>0. On suppose

(1) lim

dist(x,a)→∞

Sec_g(x) (dist(x, a))² = 0

pour touta∈M sip >2, ouSecg≤0 si 10 telle que toute solutionu∈C¹(M)de

(2) −∆pu+|∇u|^q = 0 surM

v´erifie

(3) |∇u(x)| ≤cn,p,qB^q+1−p¹ ∀x∈M.

Une des conséquences est un théorème de type Liouville.

1. Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 7350, Faculté des Sciences, 37200 Tours France. E-mail : [email protected]

2. Departamento de Matematicas, Pontifica Universidad Catolica de Chile Casilla 307, Correo 2, Santiago de Chile. E-mail : [email protected]

3. Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 7350, Faculté des Sciences, 37200 Tours France. E-mail : [email protected]

1

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2 QUASILINEAR EQUATIONS ON MANIFOLDS

Corollaire 2. Supposons que Ricc_g ≥0, q > p−1 >0 et que les hypothèses du Théorème 1 portant sur la courbure sectionnelle soient vérifiées si p 6= 2. Alors toute solutionu∈C¹(M)de (2) est constante.

Sivest une fonctionp-harmonique positive surM, la fonctionu:=−(p−1) lnv v´erifie

(4) −∆pu+|∇u|^p= 0 surM.

En utilisant le résultat du théorème 1, on en déduit

Théorème 3. Supposons quep >1 et que les hypothèses du Théorème 1 portant sur la courbure soient vérifiées. Il existe alors une constantecn,p>0telle que toute fonctionp-harmonique et positive v surM vérifie

(5) v(a)e⁻^c(n,p)Bdist^(x,a)≤v(x)≤v(a)e^c(n,p)Bdist^(x,a) ∀(a, x)∈M×M.

Quand p= 2 Cheng et Yau [1] ont montré que toute fonction harmonique positive sur une variété riemannienne complète à courbure de Ricci positive est une constante. Dans le cas des fonctions p-harmoniques positives et sous l’hypothèse de minoration uniforme de la courbure sectionnelle,Secg≥ −B², Kotschwar et Ni [4] montrent que toute fonctionp-harmonique positivevsurM vérifie l’estimation suivante,

(6) |∇v|

v ≤(p−1)B.

Notons que leur hypoth`ese impliqueRicc_g≥(1−n)B².

———————————————————————————————

Let (Mⁿ, g) be a complete, connected and non compact Riemannian manifold with Ricci curvatureRiccg and sectionnal curvatureSecg. Forp >1 we denote by

∆p the p-Laplacian defined in the metricg by

∆pu:=div

|∇u|^p⁻²∇u ,

and thus ∆2is the Laplace-Beltrami operator on M. Ifp= 2 a classical result due to Cheng and Yau [1] asserts that ifRicc_g is nonnegative, any nonnegative harmonic function v is a constant. In [4], Kotschwar et Ni obtained sharper results dealing with positivep-harmonic functions under the assumption thatSecg≥ −B². They proved that ifv is such a function, it satisfies

(1) |∇v|

v ≤(p−1)B.

Their assumption onSec_gimpliesRicc_g≥(1−n)B². They also noticed that ifp= 2 their estimate holds under the previous lower estimate on the Ricci curvature. In this note we give an extension of their result in imbedding it the more general class of quasilinear Hamilton-Jacobi type equations

(2) −∆pu+|∇u|^q= 0 onM.

Our main result is the following

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Theorem 1. Let B≥0 such thatRicc_g≥(1−n)B². Ifp >2we assume that for any a∈M

(3) lim

dist(x,a)→∞

Sec_g(x) (dist(x, a))² = 0,

and if 10 such that any solution u∈C¹(M)of (2) satisfies

(4) |∇u(x)| ≤c_n,p,qB^q+1¹^−p ∀x∈M.

A clear consequence of (3) is the following Liouville theorem

Corollary 2.AssumeRiccg≥0and that the assumptions of Theorem 1 concerning Sec_g hold ifp6= 2. Then any solution u∈C¹(M)of (2) is constant.

Ifv is a positivep-harmonic function onM, thenu:=−(p−1) lnvsatisfies

(5) −∆pu+|∇u|^p= 0 onM.

Therefore estimate (4) yields to the following result

Theorem 3. Assume that p >1 and the curvature assumptions of Theorem 1 are fulfilled. Then there exists a constant cn,p >0 such that any positive p-harmonic function v onM satisfies

(6) v(a)e⁻^c(n,p)Bdist(x,a)

≤v(x)≤v(a)e^c(n,p)Bdist(x,a)

∀(a, x)∈M×M.

Proof of Theorem 1.LetM+:={x∈M :|∇u(x)|>0}. ThenM+ is open and u∈C³(M+) since the equation is no longer degenerate. The proof is based upon the fact thatz=|∇u|²is a subsolution of an elliptic differential inequality with a superlinear absorption term (see [5] for other applications). We denote byT M the tangent bundle ofM and byh., .ithe scalar product induced by the metricg. We recall that anyC³-functionuverifies the B¨ochner-Weitzenb¨ock formula ; combined with Schwarz inequality it yields to

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1

2∆2|∇u|²= D²u

2+h∇∆2u,∇ui+Riccg(∇u,∇u)

≥ 1

n|∆2u|²+h∇∆2u,∇ui+Ricc_g(∇u,∇u),

whereD²uis the Hessian. If uis a C¹ solution of (2), thenz=|∇u|²satisfies

(8) −∆2u−p−2

2

h∇z,∇ui

z +z^q+2−p² = 0 onM+. Replacing ∆2uin (7) it follows that, for anya >0, (9)

∆2z+ (p−2)hD²z(∇u),∇ui

z ≥2a²

N z^q+2⁻^p− 1 N a²

h∇z,∇ui²

z² −(p−2) 2

|∇z|² z + (p−2)h∇z,∇ui²

z² + (q+ 2−p)z^q−p² h∇z,∇ui −(N−1)B²z.

Sincez^q−p² |h∇z,∇ui| ≤z^q+1²^−p|∇z|

√z , we can takea=a(p, q)>0 large enough so that the right-hand side of (9) is bounded from below by Cz^q+2⁻^p−D|∇z|²

z for

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someC, D >0 which depend only onpandq. We set A(v) :=−∆2v−(p−2)hD²v(∇u),∇ui

|∇u|² =−

N

X

i,j=1

aijvxixj

where theaij depend on∇uand satisfy θ|ξ|²≤

N

X

i,j=1

a_ijξ_iξ_j ≤Θ|ξ|² ∀ξ= (ξ1, ..., ξ_n)∈Rⁿ, whereθ= min{1, p−1} and Θ = max{1, p−1}. Then

(10) L^∗(z) :=A(z) +Cz^q+2⁻^p−D|∇z|²

z −(n−1)B²z≤0 in M+. The next lemma is a local estimate.

Lemma 1. Let B_R(a)⊂Mⁿ(g)be the geodesic ball of radiusR >0 and centera.

Assume that Ricc_g ≥ −(n−1)B² and either Sec_g ≥ −S² for some S² :=S_R² in BR(a)if p >2, or Secg≤0 if 10 such that the function

(11) w(x) =λ R²−r²(x)⁻q+1²−p +µ with r=r(x) =d(x, a), satisfiesL^∗(w)≥0 inBR(a), provided

(12) λ=cmaxn

(R⁴B²)^q+1−p¹ ,((1 +B+ (p−2)+S)R³)^q+1−p¹ o and

(13) µ= ((n−1)B²)^q+1−p¹ .

Proof. We recall that ∆2w=w^′′+w^′∆2rand by [6, Lemma 1]

∆2r≤(n−1)Bcoth(Br)≤ n−1

r (1 +Br). Then

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∆2w≤ 4

q+ 1−p(R²−r²)⁻^2(q+2−p)^q+1−p

2r²(q+ 3−p)

q+ 1−p + (R²−r²)(1 + (n−1) (1 +Br)

. Moverover from [3, Chap 2, p. 23]

(15) D²w=w^′′dr⊗dr+w^′D²r.

If 0≥Secg(x)≥ −S², there holds

(16) 0≤D²r≤Scoth(Sr)g≤ S

r (1 +Sr)g.

Therefore, ifp≥2 andSecg≥ −S², we get (17)

hD²w(∇u),∇ui

|∇u|² ≤ 4

q+ 1−p(R²−r²)⁻^2(q+2−p)^q+1^−p

2r²(q+ 3−p)

q+ 1−p + (R²−r²)(2 +Sr)

, while, ifp≤2 andSec_g≤0,

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(18)

hD²w(∇u),∇ui

|∇u|² ≤ 4

q+ 1−p(R²−r²)⁻^2(q+2^q+1^−p^−p)

2r²(q+ 3−p)

q+ 1−p + 2(R²−r²)

.

As a consequence

(19) A(w) =−∆w−(p−2)hD²w(∇u),∇ui

|∇u|²

≥ −kλ(R²−r²)⁻^2(q+2−p)^q+1−p (R²+ (R²−r²)Bpr) for somek=k(n, p, q), whereB_p=B+ (p−2)+S. Since

w^q+2⁻^p ≥λ^q+2⁻^p R²−r²⁻^2(q+1_q+1−p^−p)

+µ^q+2⁻^p, we have

(20)

L^∗(w)≥λ(R²−r²)⁻^2(q+2−p)^q+1−p

−k(R²+ (R²−r²)Bpr)−D 16

(q+ 1−p)²r²+Cλ^q+1⁻^p

+µ^q+2⁻^p−(n−1)B²λ R²−r²⁻_q+1−p²

−(n−1)B²µ.

We first take

(21) µ= ((n−1)B²)^q+1¹^−p.

Next we chooseλin order to have, uniformly for 0≤r < R, 2⁻¹Cλ^q+1⁻^p≥k R²+ (R²−r²)Bpr

+ 16Dr² (q+ 1−p)² and

2⁻¹Cλ^q+2⁻^p(R²−r²)⁻^2(q+2−p)^q+1−p ≥(n−1)B²λ R²−r²⁻q+1²−p. There existsc=c(n, p, q) such that, if

(22) λ=cmaxn

(R⁴B²)^q+1¹^−p,((1 +Bp)R³)^q+1¹^−po , thenL^∗(w)≥0 holds.

Lemma 2. Under the assumptions of Lemma 1, any C¹ solution of (2) in M satisfies

(23)

|∇u(x)| ≤c_n,p,qmaxn

B^q+1¹^−p,(1 +B_p)^2(q+1−p)¹ (d(x, ∂Ω))⁻^2(q+1−p)¹ o

∀x∈Ω, for every domain Ω⊂M, whereB_p=B+ (p−2)+S andS =S_d(x,∂Ω).

Proof. Assume a∈Ω, with R < d(a, ∂Ω). Letw be as in Lemma 1, then in any connected componentGof{x∈B_R(a) :z(x)−w(x)>0}we find

(24)

A(z−w) +C z^q+2⁻^p−w^q+2⁻^p

−(n−1)B²(z−w)−D |∇z|²

z −|∇w|² w

!

≤0.

By the mean value theorem and since w(a) is the minimum ofw, there holds (25) C(z^q+2⁻^p−w^q+2⁻^p)−(n−1)B²(z−w)>0,

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providedC(q+ 2−p)(w(a))^q+1⁻^p >(n−1)B². Sincew(a)> µ= (n−1)B²_q+1¹_−p and q+ 2−p > 1, this condition is fulfilled, up to replacing µ by Aµ for some A=A(p, q)>1. Ifx0∈Gis such thatz−wis maximal atx0, we derive that A(z−w) +C z^q+2⁻^p−w^q+2⁻^p

−(N−1)B²(z−w)−D |∇z|²

z −|∇w|² w

!

≤0 ifx=x0, which is a contradiction. ThusG=∅,z≤wand (23) follows.

The proof of Theorem 1 and Corollary 2 follows by taking Ω =BR(x) and letting R→ ∞.

Proof of Theorem 3.We takeq=pand assume thatvisp-harmonic and positive.

If we writev=e⁻^p−^u¹ , thenusatisfies

−∆pu+|∇u|^p= 0.

If Riccg(x) ≥ 0, uis constant by Corollary 2, and so is v. If inf{Riccg(x) : x ∈ M}= (1−n)B²<0 we apply (23) to∇u. Ifγ is a minimizing geodesic fromato x, then|γ^′(t)|= 1 and

u(x)−u(a) =

Z d(x,a)

0

d

dtu◦γ(t)dt=

Z d(x,a)

0 h∇u◦γ(t), γ^′(t)idt.

Since

|h∇u◦γ(t), γ^′(t)| ≤ |∇u◦γ(t)| ≤cn,p,pB, we obtain

(26) u(a)−c_n,p,pBdist (x, a)≤u(x)≤u(a) +c_n,p,pBdist (x, a) ∀x∈M.

Then (6) follows sinceu= (1−p) lnv.

R´ef´erences

[1] S. Y. Cheng and S.-T. Yau : Differential equations on Riemannian manifolds and their geometric applications,Comm. Pure Appl. Math. 28, 333–354 (1975).

[2] T. H. Colding and W.P. Minicozzi :An excursion into geometric analysis, Grigor’yan, Alexan- der (ed.) et al., Surveys in differential geometry. Somerville, MA : International Press,Sur- veys in Differential Geometry 9, 83-146 (2004).

[3] R.E. Greene and H. Wu/Function Theory on Manifolds Which Possess a Pole,Lecture Notes in Mathematics 699, Sringer-Verlag Berlin Heidelberg (1979).

[4] B. L. Kotschwar and L. Ni :Local gradient estimates of p-harmonic functions, 1/H-flow, and an entropy formula,Ann. Sci. ´Ecole. Norm. Sup. 42, 1–36 (2009).

[5] A. Porretta and L. V´eron :Separable p-harmonic functions in a cone and related quasilinear equations on manifolds,J. Eur. Math. Soc. 11, 1285-1305 (2009).

[6] A. Ratto, M. Rigoli and L. V´eronConformal immersion of complete Riemannian manifolds and extensions of the Schwarz lemma,Duke Math. J. 74, 223-236 (1994).