AN INDUCTION PRINCIPLE FOR THE WEIGHTED p-ENERGY MINIMALITY OF x/llxll.

(1)

HAL Id: hal-00018444

https://hal.archives-ouvertes.fr/hal-00018444

Preprint submitted on 2 Feb 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

AN INDUCTION PRINCIPLE FOR THE WEIGHTED p-ENERGY MINIMALITY OF x/llxll.

Jean-Christophe Bourgoin

To cite this version:

Jean-Christophe Bourgoin. AN INDUCTION PRINCIPLE FOR THE WEIGHTED p-ENERGY MINIMALITY OF x/llxll.. 2006. �hal-00018444�

(2)

ccsd-00018444, version 1 - 2 Feb 2006

p-ENERGY MINIMALITY OF x/kxk

JEAN-CHRISTOPHE BOURGOIN

Abstract. In this paper, we investigate minimizing properties of the mapx/kxkfrom the Euclidean unit ballBⁿto its boundarySⁿ⁻¹, for the weighted energy functionalsE_p,αⁿ (u) =R

Bⁿkxk^αk∇uk^pdx. We establish the following induction principle: if the map _kxk^x :Bⁿ⁺¹→Sⁿminimizes Eⁿ⁺¹p,α among the maps u : Bⁿ⁺¹ → Sⁿ satisfying u(x) = x on Sⁿ, then the map _kyk^y : Bⁿ → Sⁿ⁻¹ minimizes Eⁿp,α+1 among the maps v:Bⁿ→Sⁿ⁻¹satisfyingv(y) =yonSⁿ⁻¹.

This result enables us to enlarge the range of values ofp andαfor whichx/kxkminimizesEⁿp,α.

1. Introduction and statement of main results

Let Bⁿ be the unit Euclidean ball of Rⁿ and Sⁿ⁻¹ its bourdary. For any couple (α, p) of real numbers, with α ≥ 0 and p ≥ 1, we define the r^α-weighted p-energy functional of a mapu :Bⁿ→Sⁿ⁻¹ by

E_p,αⁿ (u) = Z

Bⁿkxk^αk∇uk^pdx.

This functional is nothing but the p-energy functional associated with the Riemannian metric r^α(n−p)g_euc on the ballBⁿ, where g_euc is the Euclidean metric, the sphere Sⁿ⁻¹ being endowed with its standard metric. The functional E_p,αⁿ is to be considered on the Sobolev space

W_α^1,p(Bⁿ,Sⁿ⁻¹) ={u∈W^1,p((Bⁿr^α(n−p)g_euc),Rⁿ);kuk= 1a.e}. The question is to know which map minimizes E_p,α among the maps in W_α^1,p(Bⁿ,Sⁿ⁻¹) satisfyingu(x) =x on Sⁿ⁻¹.

Following the arguments of Hildebrant, Kaul and Widman [11], the map x/kxk is a natural candidate to be the minimizer of E_p,αⁿ , forp∈[1, n+α) (notice that x/kxk ∈Wα^1,p(Bⁿ,Sⁿ⁻¹) if and only ifp < n+α).

The minimality ofx/kxk was first established for the 2-energy functional Eⁿ₂ := E_2,0ⁿ by J¨ager and Kaul [14] in dimension n ≥ 7, then by Brezis, Coron and Lieb [2] in dimension 3. Coron and Gulliver [5] actually proved the minimality of x/kxk for the p-energy functional E_pⁿ := E_p,0ⁿ for any integer p ∈ {1,· · · , n−1} and any dimension n≥3. An alternative proof using the null Lagrangian method (or calibration method) of this last result was obtained by Lin [15] and Avellaneda and Lin [1].

Hardt and Lin [7, 8] and Hardt, Lin and Wang [9, 10] developed the study of the singularities ofp-harmonic andp-minimizing maps and obtained results extending those of Schoen and Uhlenbelck [16, 17]. A consequence of their results is the minimality of x/kxk for E_pⁿ for any p ∈ (n−1, n).

1

(3)

2 JEAN-CHRISTOPHE BOURGOIN

Finally, Hong [12] and Wang [18] have proved independently the minimality of x/kxk forE_pⁿ in dimensionn≥7 for anyp≤n−2√

n−1.

In order to close the question of the E_pⁿ-minimality of x/kxk for the remaining values of p, Hong [13] suggested a new idea. Indeed, he observed that, for any p ∈ (2, n), the minimality of x/kxk for Eⁿ_p follows from the minimality of x/kxk for the r^2−p-weighted 2-energy E_2,2−pⁿ . Unfortunately, we have proved in a previous paper [3], the existence of an interval of values of p∈(1, n−1) for which the mapx/kxk fails to be a minimizer of E_2,2−pⁿ . Nevertheless, we proved thatx/kxkminimizes ther^α-weightedp-energyEⁿ_p,α for any α ≥0 and any integer p ≤n−1. Actually, we obtained in [4] the minimality of x/kxk for more general weighted p-energy functionals of the form E_p,fⁿ (u) = R

Bⁿf(kxk)k∇uk^pdx, where p ≤ n−1 is an integer and f : [0,1]→R⁺ is a continuous non-decreasing function.

Our aim in this paper is to show that the minimizing properties ofx/kxk in dimension n and n+ 1 are not independent. Indeed, we will prove that, for anyp∈[1, n+α+ 1), ifx/kxkminimizes ther^α-weightedp-energyE_p,αⁿ⁺¹ in dimensionn+1, then it also minimizes ther^α+1-weightedp-energyE_p,α+1ⁿ in dimension n.

Theorem 1. Let n ≥ 2 be an integer, α ≥ 0 be a real number and p ∈ [1, n+α+ 1). If the map _kxk^x : Bⁿ⁺¹ 7→ Sⁿ minimizes the r^α-weighted p- energy E_p,αⁿ⁺¹ among the maps u ∈ W_α^1,p(Bⁿ⁺¹,Sⁿ) satisfying u(x) = x on Sⁿ, then the map _kyk^y : Bⁿ 7→ Sⁿ⁻¹ minimizes the r^α+1-weighted p-energy Eⁿ_p,α+1 among the maps v∈W_α+1^1,p (Bⁿ,Sⁿ⁻¹) satisfyingv(y) =y on Sⁿ⁻¹.

The proof of this theorem relies on a construction which associates to each mapu:Bⁿ→Sⁿ⁻¹such thatu(y) =yonSⁿ⁻¹, a mapu :Bⁿ⁺¹ →Sⁿ such that u(x) =u(x) in Bⁿ× {0} and u(x) = x on the unit sphere Sⁿ, in such a way that, if u₀ is the map defined inBⁿ by u₀(y) = _kyk^y , then u₀ is exactly the map defined on Bⁿ⁺¹ by u₀(x) = _kxk^x . The energy E_p,αⁿ⁺¹(u) of u is estimated above in terms of the energy E_p,α+1ⁿ (u) ofu and the equality holds in the estimate for the map u₀ = _kyk^y .

Thanks to Theorem 1 and the minimality results mentioned above, we deduce the following :

Corollary 1. The mapy/kykminimizesE_p,αⁿ among the maps inWα^1,p(Bⁿ,Sⁿ⁻¹) which coincide with y/kyk on Sⁿ⁻¹ in the following cases :

i) α∈Nand p∈(n+α−1, n+α),

ii) α ≥β and p is an integer in {1,· · · , n+β−1}, for any real number β ≥0.

iii) α∈N, n+α≥7 and p≤n+α−2√

n+α−1,

1.1. Construction ofuand proof of Theorem 1.1. LetBⁿ⁺¹andSⁿbe the unit open ball and the unit sphere of Rⁿ⁺¹ and letBⁿ andSⁿ⁻¹ be the unit open ball and the unit sphere of Rⁿ that we identify with the subspace Rⁿ=Rⁿ× {0}inRⁿ⁺¹. Moreover, we writex= (x₁,· · · , x_n, x_n+1) a vector of Rⁿ⁺¹,y = (y1,· · · , yn) a vector ofRⁿ, h., .i the standard metric of Rⁿ⁺¹ and (e₁,· · ·, e_n, e_n+1) the standard basis ofRⁿ⁺¹.

(4)

Let Π_n be the projection defined by : Π_n:Bⁿ⁺¹ −→ Bⁿ

(x₁,· · · , x_n+1) −→ (x₁,· · ·, x_n,0) = (x₁,· · · , x_n).

Consider the map ϕ_n defined onBⁿ\Re_n+1 byϕ_n(x) = _kΠ^Πⁿ^(x)

n(x)k. We define the map udefined by

u:Bⁿ⁺¹ −→ Bⁿ x −→ he_n+1, x

kxkie_n+1+hϕ_n(x), x

kxkiu(kxkϕ_n(x)).

Lemma 1. For any x∈Bⁿ⁺¹,k∇u(x)k² = _kxk¹2 +k∇u(kxkϕ_n(x))k². Proof. For anyi∈ {1,· · · , n}, we have,

du(x).e_i = hΠ_n(x), e_ii kxk

1

kΠ_n(x)k −kΠ_n(x)k kxk²

u(kxkϕ_n(x))

−he_n+1, xihx, e_ii kxk³ e_n+1

+du(kxkϕ_n(x)). hΠn(x), eiiΠn(x)

kxk² +e_i−hΠn(x), eiiΠn(x) kΠ_n(x)k²

!

and,

du(x).e_n+1 = e_n+1

kxk − he_n+1, xi²

kxk³ e_n+1−he_n+1, xi

kxk³ kΠ_n(x)ku(kxkϕ_n(x)) +he_n+1, xi

kxk² du(kxkϕ_n(x)).Π_n(x).

Since ku(x)k² = 1, one has hdu(x).h, u(x)i = 0 for any h∈ Rⁿ. Hence, for any i∈ {1,· · · , n}, we have,

kdu(x).e_ik² = hΠ_n(x), e_ii² kxk²

1

kΠn(x)k −kΠ_n(x)k kxk²

2

+ he_n+1, xi²hx, e_ii² kxk⁶

+

du(kxkϕ_n(x)).

hΠ_n(x), e_ii 1

kxk² − 1 kΠ_n(x)k²

Π_n(x) +e_i

2.

kdu(x).e_n+1k² = 1 kxk²

1−he_n+1, xi² kxk²

2

+he_n+1, xi²

kxk⁶ kΠ_n(x)k² +hen+1, xi²

kxk⁴ kdu(kxkϕ_n(x)).Π_n(x)k². Finally, we have,

k∇u(x)k² = 1

kxk² +k∇u(kxkϕ_n(x))k².

Letx_n+1be a real number in (0,1), consider the setA_xn+1 = (x_n+1e_n+1+ e^⊥_n+1)∩Bⁿ⁺¹\Re_n+1, where e^⊥_n+1 is the orthogonal subspace to Re_n+1 for

(5)

h., .i. Let θbe the map :

θ:Axn+1 −→ Cxn+1 ={y∈Rⁿ;kyk>|xn+1|}

x= (x₁,· · · , x_n+1) −→ kxkϕ_n(x) =y = (y₁,· · ·, y_n).

Lemma 2. For any y∈Rⁿ, the Jacobian determinant of θ⁻¹ is : Jac(θ⁻¹)(y) = kyk²−x²_n+1ⁿ⁻²₂

kykⁿ⁻² . Proof For any i∈ {1,· · ·, n}and for any x∈A_x_n+1,

dθ(x).e_i = 1

kxk − kxk kΠ_n(x)k²

hΠ_n(x), e_ii

kΠ_n(x)k Π_n(x) + kxk kΠ_n(x)ke_i. Let us set, for any i∈ {1,· · · , n},

λ_i= 1

hΠ_n(x), e_ii

kΠ_nk and α= kxk kΠ_n(x)k. Hence, for any i∈ {1,· · ·, n},

dθ(x).e_i=λ_iΠ_n(x) +αe_i. Then, for any x∈A_x_n+1,

Jac(θ)(x) = det(λ₁Π_n(x) +αe₁, λ₂Π_n(x) +αe₂,· · · , λ_nΠ_n(x) +αe_n)

=

n

X

i=1

det(αe₁,· · · , αe_i−1, λ_iΠ_n(x), αe_i+1,· · · , αe_n) +αⁿdet(e₁,· · · , e_n)

=

n

X

i=1

αⁿ⁻¹λ_idet(e₁,· · · , e_i−1,Π_n(x), e_i+1,· · ·, e_n) +αⁿ,

where det is the determinant in the basis (e₁,· · ·, e_n+1).

Jac(θ)(x) =

n

X

i=1

αⁿ⁻¹ 1

hΠ_n(x), e_ii² kΠ_n(x)k +αⁿ

= αⁿ⁻¹ 1

kxk− kxk kΠn(x)k²

kΠ_n(x)k²

kΠn(x)k +αⁿ =αⁿ⁻². Therefore, we have

Jac(θ)(x) = kxkⁿ⁻²

kΠ_n(x)kⁿ⁻² = kykⁿ⁻² kyk²−x²_n+1ⁿ⁻₂²

.

We deduce that,

Jac(θ⁻¹)(y) = kyk²−x²_n+1ⁿ⁻²₂ kykⁿ⁻² .

(6)

Lemma 3. For any x∈Bⁿ⁺¹, we have, Z

Bⁿ⁺¹kxk^αk∇u(x)k^pdx₁· · ·dx_n+1 ≤ n^p/2−1 Z

Bⁿ⁺¹

1

kxk^p−αdx₁· · ·dx_n+1 +2(1−1/n)^1−p/2W_n−1

Z

Bⁿkyk^α+1k∇u(y)k^pdy, where W_n−1=R^π₂

0 (cos(γ))ⁿ⁻¹dγ.

Proof. Writing _kxk¹2+k∇u(kxkϕ_n(x))k² = _n¹(n_kxk¹2)+(1−¹_n)( ¹

1−¹

n

k∇u(kxkϕ_n(x))k²) and using that x→x^p² is a convex map, for anyp∈[1, n+α+ 1), we have,

k∇u(x)k^p = 1

kxk² +k∇u(kxkϕ_n(x))k² p/2

≤ n^p/2−1 1

kxk^p + (1−1/n)^1−p/2k∇u(kxkϕ_n(x))k^p. Hence,

Z

Bⁿ⁺¹kxk^α 1

kxk^pdx₁· · ·dx_n+1 +(1−1/n)^1−p/2

Z

Bⁿ⁺¹kxk^αk∇u(kxkϕ_n(x))k^pdx₁· · ·dx_n+1

≤ n^p/2−1 Z

Bⁿ⁺¹kxk^α 1

kxk^pdx₁· · ·dx_n+1 + 2(1−1/n)^1−p/2

Z 1 0

dxn+1

Z

Bⁿkxk^αk∇u(kxkϕn(x))k^pdx1· · ·dxn. Using the change of variables y=θ(x) and Lemma 1.2 we get,

Z

Bⁿ⁺¹

1

kxk^p−αdx₁· · ·dx_n+1

+ 2(1−1/n)^1−p/2 Z ₁

0

dx_n+1 Z

Cⁿ⁺¹

kyk²−x²_n+1ⁿ⁻²₂

kykⁿ⁻² kyk^αk∇u(y)k^pdy₁· · ·dy_n, Z

Bⁿ⁺¹

1

kxk^p−αdx₁· · ·dx_n+1

+ 2(1−1/n)^1−p/2 Z

Bⁿkyk^αk∇u(y)k^p



 Z kyk

0

kyk²−x²_n+1ⁿ⁻²₂ kykⁿ⁻² dxn+1



dy.

But we have, Z kyk

0

kyk²−x²_n+1 kyk²

ⁿ⁻²2

dx_n+1 = Z kyk

0

1− x_n+1

kyk

2!ⁿ⁻²₂ dx_n+1

= Z 1

0 kyk(1−t²)ⁿ⁻²² dt=kyk Z π/2

0

(cosγ)ⁿ⁻¹dγ.

(7)

Let us set W_n−1 =R_π/2

0 (cosγ)ⁿ⁻¹dγ. Then, we have the inequality, Z

Bⁿ⁺¹

1

kxk^p−αdx₁· · ·dx_n+1 + 2(1−1/n)^1−p/2W_n−1

Z

Bⁿkyk^α+1k∇u(y)k^pdy.

Lemma 4. Let Γ be the Gamma function defined by, Γ(x) =

Z _+∞

0

t^x−1e^−tdt.

We have,

Wn−1

Γ(ⁿ⁺¹₂ ) Γ(ⁿ₂) =

√π 2 .

Proof From the equality Γ(x+ 1) =xΓ(x) for anyx∈(0,+∞), we have, ifn= 2l, where l∈N,

Γ(2l+ 1

2 ) = 2l−1 2

2l−3 2 · · ·3

2 1 2Γ(1

2) and

Γ(2l

2) = 2l−2 2

2l−4 2 · · ·2

2Γ(1).

Moreover we have,

W_2l−1 = (2l−2)(2l−4)· · ·2 (2l−1)(2l−3)· · ·3.1 then,

W_2l−1Γ(^2l+1₂ ) Γ(^2l₂) =

√π 2 . If n= 2l+ 1 where l∈N, we obtain,

Γ(2l+ 2 2 ) = 2l

2 2l−2

2 · · ·2 2Γ(1), Γ(2l+ 1

2 ) = 2l−1 2 · · ·3

2 1 2Γ(1

2) and

W_2l= (2l−1)(2l−3)· · ·3.1 (2l)(2l−2)· · ·2 .π

2. We deduce that

W_2lΓ(^2l+2₂ ) Γ(^2l+1₂ ) =

√π

2 .

Proof of Theorem 1.1. Suppose that u₀ is a minimizer map of E_p,α. Since the measure of Sⁿ is |Sⁿ| = ^2π

n+1 2

Γ(ⁿ⁺¹₂ ) and since k∇u0k(x) = ⁿ

p 2

kxk, by

(8)

Lemma 1.3, for any map u∈W_α+1^1,p (Bⁿ,Sⁿ⁻¹) satisfying u(y) =y on Sⁿ⁻¹, we have,

n^p/2 n+ 1 +α−p

2πⁿ⁺¹² Γ(ⁿ⁺¹₂ ) =

Z

Bⁿ⁺¹kxk^αk∇u0(x)k^pdx1· · ·dxn+1

≤ Z

Bⁿ⁺¹kxk^αk∇u(x)k^pdx₁· · ·dx_n+1

≤ n^p/2−1 Z

Bⁿ⁺¹

1

kxk^p−αdx1· · ·dxn+1

+2(1−1/n)^1−p/2Wn−1

Z

Bⁿkyk^α+1∇u(y)k^pdy

≤ 1 n

n^p/2 n+1+α−p

2πⁿ⁺¹² Γ(ⁿ⁺¹₂ ) + 2W_n−1(1− 1

n)( n n−1)^p/2

Z

Bⁿkyk^α+1k∇u(y)k^pdy.

Then we have, 2Wn−1

1−1

n

n n−1

p/2R

Bⁿkyk^α+1k∇u(y)k^pdy R

Bⁿkyk^α+1k∇_ky)k^y k^pdy ≥ n^p/2 n+1+α−p

2πⁿ⁺¹² Γ(ⁿ⁺¹₂ )

×n+1+α−p (n−1)^p/2

Γ(ⁿ₂) 2πⁿ²

1−1

n

By Lemma 1.4 we finally get, Z

Bⁿkyk^α+1k∇u(y)k^pdy≥ Z

Bⁿkyk^α+1k∇ y

kykk^pdy

References

[1] M. Avellaneda, F.-H. Lin, Null-Lagrangians and minimizingR

|∇u|^p,C. R. Acad.

Sci. Paris,306(1988), 355-358.

[2] H. Brezis, J.-M. Coron, E.-H. Lieb, Harmonic Maps with Defects,Commun. Math.

Phys.,107(1986), 649-705.

[3] J.-C. Bourgoin, The minimality of the mapx/kxkfor weighted energy,Calculus of Variation and P.D.E’s, (to appear in number526).

[4] J.-C. Bourgoin, On the minimality of thep-harmonic map for weignted energy, submit for publication in Ann. and global analysis geometry,

[5] J.-M. Coron, R. Gulliver, Minimizing p-harmonic maps into spheres, J. reine angew. Math.,401(1989), 82-100.

[6] B. Chen, R. Hardt, Prescribing singularities for p-harmonic mappings, Indiana University Math. J.,44(1995), 575-601.

[7] R. Hardt, F.-H. Lin, Mapping minimizing the L^p norm of the gradient, Comm.

P.A.M., 15(1987), 555-588.

[8] R. Hardt, F.-H. Lin, Singularities forp-energy minimizing unit vectorfields on pla- nar domains ,Calculus of Variations and Partial Differential Equations,3(1995), 311-341.

[9] R. Hardt, F.-H. Lin, C.-Y. Wang, Thep-energy minimality of _kxk^x , Communica- tions in analysis and geometry,6(1998), 141-152.

(9)

[10] R. Hardt, F.-H. Lin, C.-Y. Wang, Singularities ofp-Energy Minimizing Maps, , Comm. P.A.M., 50(1997), 399-447.

[11] S. Hildebrandt, H. Kaul, K.-O. Wildman, An existence theorem for harmonic mappings of Riemannian manifold,Acta Math.,138(1977), 1-16.

[12] M.-C. Hong,, On the Jager-Kaul theorem concerning harmonic maps, Ann.

Inst. Poincar, Analyse non-linaire,17(2000), 35-46.

[13] M.-C. Hong, On the minimality of thep-harmonic map _kxk^x :Bⁿ →Sⁿ⁻¹,Calc.

Var.,13(2001), 459-468.

[14] W. Jager, H. Kaul, Rotationally symmetric harmonic maps from a ball into a sphere and the regularity problem for weak solutions of elliptic systems, J.Reine Angew. Math.,343(1983), 146-161.

[15] F.-H. Lin, Une remarque sur l’application x/kxk, C.R. Acad. Sci. Paris 305(1987), 529-531.

[16] R. Shoen, K. Uhlenbeck, A regularity theory for harmonic mapsJ. Differential Geom.. 12(1982), 307-335.

[17] R. Shoen, K. Uhlenbeck, Boundary theory and the Dirichlet problem for harmonic maps, J. Differential Geom., 18(1983), 253-268.

[18] C.Wang, Minimality and perturbation of singularities for certain p-harmonic maps., Indania Univ. Math.J.,47(1998), 725-740.

Laboratoire de Mathématiques et Physique Théorique, UMR CNRS 6083, Université de Tours, Parc de Grandmont, F-37200 Tours France