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Comptes Rendus
Mathématique
Chang-Yu Guo and Chang-Lin Xiang
Conservation law of harmonic mappings in supercritical dimensions
Volume 362 (2024), p. 769-773
Online since: 17 September 2024
https://doi.org/10.5802/crmath.592
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Creative Commons Attribution 4.0 International License.
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C EN T R E
MER S ENN E
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Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2024, Vol. 362, p. 769-773
https://doi.org/10.5802/crmath.592
Research article / Article de recherche
Partial differential equations / Equations aux dérivées partielles
Conservation law of harmonic mappings in
supercritical dimensions
Loi de conservation des applications harmoniques en
dimensions surcritiques
Chang-Yu Guo a and Chang-Lin Xiang ∗,b
a Research Center for Mathematics and Interdisciplinary Sciences, Shandong
University, 266237, Qingdao and Frontiers Science Center for Nonlinear Expectations,
Ministry of Education, P. R. China
b Three Gorges Mathematical Research Center, China Three Gorges University,
443002, Yichang, P. R. China
E-mails: changyu.guo@sdu.edu.cn (C.-Y. Guo), changlin.xiang@ctgu.edu.cn
(C.-L. Xiang)
Abstract. In this short note, we provide a partial extension of Rivière’s convervation law in higher dimensions
under certain Lorentz integrability condition for the connection matrix. As an application, we obtain a
conservation law for weakly harmonic mappings around regular points in supercritical dimensions.
Résumé. Dans cette courte note, nous fournissons une extension partielle de la loi de conservation obtenue
par Rivière’s en dimensions supérieures, sous certaines conditions d’intégrabilité de Lorentz pour la matrice
de connexion. Comme application, nous obtenons une loi de conservation pour les applications faiblement
harmoniques autour de points réguliers en dimensions supercritiques.
2020 Mathematics Subject Classification. 58E20, 35J60.
Funding. C.-Y. Guo is supported by the Young Scientist Program of the Ministry of Science and Technology of
China (No. 2021YFA1002200), the National Natural Science Foundation of China (No. 12101362), the Taishan
Scholar Project and the Natural Science Foundation of Shandong Province (No. ZR2022YQ01, ZR2021QA003).
C.-L. Xiang is financially supported by the National Natural Science Foundation of China (No. 12271296).
Manuscript received 20 October 2023, accepted 20 November 2023.
1. Introduction and main result
Let B n ⊂ Rn (n ≥ 2) be the unit open ball centered at the origin and N k a closed k-dimensional
C 2-Riemannian manifold which is isometrically embedded intoRm . A weakly harmonic mapping
from B n into N is defined as a mapping u ∈W 1,2(B n , N ), which is a critical point of the Dirichlet
energy
∫
B n |∇u|2 with respect to the outer variation. Its Euler-Lagrange equation can be written
as
−∆u = A(u)(∇u,∇u), (1)
∗Corresponding author
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
770 Chang-Yu Guo and Chang-Lin Xiang
where A is the second foundamental form of N ,→ Rm . A basic problem on the theory of weakly
harmonic mappings is to find the optimal regularity.
In the case when N = Sm−1 ⊂ Rm is the standard Euclidean sphere, several authors [2, 8, 9]
have independently discovered that the following system (consisting of divergence-free vector
fields)
div
(
ui∇u j −u j∇ui
)
= 0, for all 1 ≤ i , j ≤ m, (2)
are indeed equivalent with (1). Since this system has a nice interpretation by Noether’s conserva-
tion law, equation (2) is called the conservation law of (sphere-valued) harmonic mappings. This
conservation law is very useful in investigating the regularity and compactness theory of (sphere-
valued) harmonic mappings; see e.g. Hélein [4] and the references therein fore more information.
In view of the above discovery and applications, Rivière asked, in his seminar work [6, p. 4],
the following fundamental question: how to write the harmonic mapping equations (1) into
divergence form for all dimensions n ≥ 2 and for all closed manifolds N ? He succeeded to do
this in [6] when n = 2. The idea is to write (1) in a more general form
−∆u =Ω ·∇u (3)
for some Ω ∈ L2(B n , som ⊗Rn). Indeed, suppose u is a smooth harmonic mapping from B n
into N , and let {νI = (ν1I , · · · , νmI )}m−kI=1 (I = 1, · · · , m − k) be an orthonormal frame of T ⊥N in a
neighborhood of u(x). Then
A(∇u,∇u) =−∑
I
〈
∂αu,∇∂αuνI
〉
νI =−
∑
I
〈
∂αu,∂α(νI (u))
〉
νI .
Note that 〈∂αu,νI (u)〉 = 0 for all 1 ≤α≤ n and I ≤ m −k. Thus, in coordinates, there holds
−∆ui =− ∑
j ,l ,I
∂αu
l
(
∂ jν
l
I (u)∂αu
j
)
νiI (u) =Ωij (u,∇u) ·∇u j , 1 ≤ i ≤ m,
where
Ωij (u,∇u) =
∑
l ,I
(
νiI (u)∂ jν
l
I (u)−ν jI (u)∂iνlI (u)
)
∇ul
is antisymmetric with respect to (i , j ) and grows as O(|∇u|). Hence Ω ∈ L2(B n , som ⊗Rn).
Working with equation (3), Rivière [6] proposed the following theorem concerning conserva-
tion law.
Theorem 1 ([6, Theorem I.3]). Fix m ∈N and Ω= (Ωij )1≤ i , j ≤m ∈ L2(B n , som ⊗∧1Rn). If there are
A ∈ L∞(B n , Mm(R))∩W 1,2 and B ∈W 1,2(B n , Mm(R)⊗∧2Rn) satisfying
dΩA := d A− AΩ=−d∗B , (4)
then every solution to (3) on B n satisfies the following conservation law
d
(∗Adu + (−1)n−1(∗B)∧du)= 0. (5)
In coordiantes, (4) means d Aij −
∑m
k=1 A
i
kΩ
k
j =−d∗B ij for all i , j ∈ {1, . . . , m} and (5) reads as
∂
∂xα
(
Aij
∂u j
∂xα
+ (−1)n−1B ij αβ
∂u j
∂xβ
)
= 0,
where the Einstein summation convention is used.
Thus according to Theorem 1, in order to find the conservation law for harmonic mappings,
it suffices to prove the existence of A,B for a given Ω ∈ L2(B n , som ⊗Rn). In the case n = 2,
Rivière [6, Theorem I.4] succeeded to prove the existence of A,B under a smallness assumption
on ∥Ω∥L2 . However, as noticed in [7], his proof does not work in higher dimensions under the
natural Morrey integrability assumption Ω ∈ M 2,n−2 due to the lack of Wente’s lemma (see [3, 7]
for more detailed dicussions).
Chang-Yu Guo and Chang-Lin Xiang 771
Motivated by this challenging problem, in this short note, we provide a partial solution in
the case n ≥ 3. More precisely, we obtain the following result, which can be viewed as a higher
dimensional extension of Rivière [6, Theorem 1.4].
Theorem 2. For any n,m ≥ 2, there exist constants ϵ = ϵ(m,n),C = C (m,n) > 0 satisfying the
following property. Suppose Ω ∈ Ln,2(B n , som ⊗∧1Rn) with
∥Ω∥Ln,2(B n ) ≤ ϵ.
Then there exist A ∈ L∞∩W 1,n,1(B n ,Glm) and B ∈W 1,n,2(B n , Mm ⊗∧2Rn) such that
d A− AΩ=−d∗B. (6)
Moreover, we have
∥dist(A,SOm)∥L∞(B n ) +∥d A∥Ln,1(B n ) +∥dB∥Ln,2(B n ) ≤C∥Ω∥Ln,2(B n ).
In the above theorem, Lp,q and W 1,p,q denote Lorentz spaces and Lorentz–Sobolev spaces
respectively, see e.g. [1, 5]. We would like to point out that the Ln,2-integrability condition on Ω
is somehow strong, as Ln,2 ⊊ Ln ⊊ Ln,∞ ⊊ M 2,n−2 when n ≥ 3. On the other hand, it seems to be
very difficult to refine Rivière’s technique further so that we may relax Ln,2, for example, to Ln,∞.
As that of Rivière [6], an application of Theorem 2 together with the harmonic mapping
equation in the form (3) leads to
Corollary 3. Let u ∈ W 1,2(B n , N ) be a weakly harmonic mapping. If u is smooth in Br (x0) ⊂ B n
for some r > 0 such that ∥∇u∥L2(Br (x0)) ≪ ϵ, then there exist A ∈ C∞(Br (x0),Glm(R)) and B ∈
C∞(Br (x0), Mm(R)⊗∧2Rn) such that u satisfies the conservation law (5) of Rivière in Br (x0).
We do not know how to extend Corollary 3 to neighborhoods around singular points of weakly
harmonic mappings in supercritical dimensions and refer to [3] for more discussions about the
conservation law.
2. Proof of Theorem 2
The proof of Theorem 2 follows closely that of Rivière [6, Theorem 1.4] with minor modifications.
Proof.
Step 1. here exist P ∈W 1,n,2(B n ,SOm) and ξ ∈W 1,n,2(B n , som ⊗∧2Rn) such that
P−1dP +P−1ΩP = d∗ξ in B n ,
and
∥dP∥Ln,2(B n ) +∥dξ∥Ln,2(B n ) ≤C∥Ω∥Ln,2(B n ) ≤Cϵ. (7)
The proof for this is completely similar to that of [6, Lemma A.3] or [7, Lemma 3.1], whereas
the idea dates back to Uhlenbeck [10]. Thus we omit it here.
Step 2. o continue, we first extend Ω, P,ξ to the whole space Rn with compact support in a
bounded way. By introducing Ã = AP , to find (A,B) in problem (6) is equivalent to find (Ã,B)
in the problem
d Ã = Ãd∗ξ+d∗BP. (8)
Here and hereafter, all the equations are considered in the whole space Rn . This leads us to solve
the second order problem{
−∆Ã = d∗ (Ãd∗ξ+d∗BP)=∗(d Ã∧d (∗ξ)+d (∗B)∧dP)
dd∗B = d ((d Ã− Ãd∗ξ)P−1)= d Ã∧dP−1 +d (Ãd∗ξP−1) (9)
772 Chang-Yu Guo and Chang-Lin Xiang
where we omitted the precise constant coefficients before each term. Set Ã = Â+ id. Then we will
try to find (Â,B) by solving the equations
−∆Â =∗(d Â∧d (∗ξ)+d (∗B)∧dP) ,
−∆B = d Â∧dP−1 +∗d∗ (Âd(∗ξ)P−1 +d(∗ξ)P−1) ,
dB = 0.
(10)
Step 3. olve problem (10) by a fixed point argument.
Consider in Rn the problem
−∆A =∗ (d a ∧d (∗ξ)+d (∗b)∧dP ) ,
−∆B = d a ∧dP−1 +∗d∗ (ad(∗ξ)P−1 +d(∗ξ)P−1) ,
dB = 0.
(11)
Introduce the Banach space
X= {(a,b) : a ∈ L∞∩W 1,n,2 (Rn ,Glm) ,b ∈W 1,n,2 (Rn , Mm ⊗∧2Rn)}
with the norm
∥(a,b)∥X = ∥a∥∞+∥d a∥Ln,2 +∥db∥Ln,2 .
For any (a,b) ∈X, there exists a unique (A,B) ∈X satisfying (11). Set
T (a,b) = (A,B).
We prove in below that T is a contraction mapping on X1 = {x ∈X : ∥x∥X ≤ 1}.
First note that by Hölder’s inequality in Lorentz spaces, we have
d a ∧d ∗ξ,d a ∧dP−1 ∈ L n2 ,1.
Hence from the first equation of (11) we derive A ∈W 2,n/2,1(Rn) ⊂C (Rn) with estimate
∥A∥∞ ≤Cn∥d A∥Ln,1 ≤C∥dξ∥Ln,2∥d a∥Ln,2 +C∥dP∥Ln,2∥db∥Ln,2 .
From the last two equations of (11), we derive
∥∇B∥Ln,2 ≤
∥∥dP−1∥∥Ln,2 ∥a∥∞+C (1+∥a∥∞)∥dξ∥Ln,2 .
Thus (7) implies that
∥(A,B)∥X ≤C∥Ω∥Ln,2 (∥(a,b)∥X+1) .
Similarly, for any (ai ,bi ) ∈X, i = 1,2, we have
∥T (a1,b1)−T (a2,b2)∥X ≤Cϵ∥(a1,b1)− (a2,b2)∥X .
Thus, by choosing ϵ ≪ 1 such that C∥Ω∥Ln,2(Rn ) ≤ Cϵ ≤ 1/2, we find that T is a contraction
mapping on X1 = {x ∈ X : ∥x∥X ≤ 1}. Consequently, by the standard fixed point theorem, there
exists a unique (Â,B) ∈X1 such that
T (Â,B) = (Â,B).
That is, (Â,B) solves problem (10). Moreover,∥∥(Â,B)∥∥
X
≤Cϵ(∥∥(Â,B)∥∥
X
+1)≤ 2Cϵ.
Step 4. ow let Ã = Â+ id. Then (Ã,B) solves problem (9) with∥∥Ã− id∥∥∞+∥∥d Ã∥∥Ln,2 +∥dB∥Ln,2 ≤C∥Ω∥Ln,2 .
We need to verify that (Ã,B) solves (8). By the first equation of (9) and the nonlinear Hodge theory,
there exists C ∈W 1,n,2(B n , Mm ⊗∧2Rn) with dC = 0 such that
d Ã− (Ãd∗ξ+d∗BP)= d∗C
Chang-Yu Guo and Chang-Lin Xiang 773
Then using the second equation of (9) we find that d(d∗C P−1) = 0. This gives d∗C P−1 = dD for
some D ∈W 1,n,2, with
∥dD∥Ln,2 ≤C∥∇C∥Ln,2 ≤C∥∇C∥Ln,1 .
Hence
∆C = dD ∧dP−1,
from which it follows that
∥∇C∥Ln,1 ≤Cϵ∥dD∥Ln,2 .
Thus, combining the above two estimates together gives
∥∇C∥Ln,1 ≤Cϵ∥∇C∥Ln,1 .
By choosing ϵ sufficiently small, we obtain C ≡ 0. The proof of Theorem 2 is complete. □
References
[1] R. A. Adams and J. J. F. Fournier, Sobolev spaces, 2nd edition, Academic Press Inc., 2003.
[2] Y. Chen, “The weak solutions to the evolution problems of harmonic maps”, Math. Z. 201
(1989), no. 1, pp. 69–74.
[3] C.-Y. Guo and C.-L. Xiang, “The conservation law approach in geometric PDEs”, in Surveys
in Geometric Analysis, Science Press, Beijing, 2023, pp. 51–63.
[4] F. Hélein, Harmonic maps, conservation laws and moving frames, Cambridge University
Press, 2002.
[5] R. O’Neil, “Convolution operators and L(p, q) spaces”, Duke Math. J. 30 (1963), pp. 129–
142.
[6] T. Rivière, “Conservation laws for conformally invariant variational problems”, Invent.
Math. 168 (2007), pp. 1–22.
[7] T. Rivière and M. Struwe, “Partial regularity for harmonic maps and related problems”,
Commun. Pure Appl. Math. 61 (2008), pp. 451–463.
[8] J. Rubinstein, P. Sternberg and J. B. Keller, “Reaction-diffusion processes and evolution to
harmonic maps”, SIAM J. Appl. Math. 49 (1989), pp. 1722–1733.
[9] J. Shatah, “Weak solutions and developments of singularities of the SU(2) σ-model”, Com-
mun. Pure Appl. Math. 41 (1988), pp. 459–469.
[10] K. K. Uhlenbeck, “Connections with Lp bounds on curvature”, Commun. Math. Phys. 83
(1982), pp. 31–42.


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