Comptes Rendus
Articles du volume en cours
Article de recherche - Equations aux dérivées partielles
Conservation law of harmonic mappings in supercritical dimensions
[Loi de conservation des applications harmoniques en dimensions surcritiques]
Chang-Yu Guo1 ; Chang-Lin Xiang2 
1 Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, 266237, Qingdao and Frontiers Science Center for Nonlinear Expectations, Ministry of Education, P. R. China
2 Three Gorges Mathematical Research Center, China Three Gorges University, 443002, Yichang, P. R. China
Comptes Rendus. Mathématique, Volume 362 (2024), pp. 769-773.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.592
Classification : 58E20, 35J60

Chang-Yu Guo 1 ; Chang-Lin Xiang 2

1 Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, 266237, Qingdao and Frontiers Science Center for Nonlinear Expectations, Ministry of Education, P. R. China
2 Three Gorges Mathematical Research Center, China Three Gorges University, 443002, Yichang, P. R. China
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{CRMATH_2024__362_G7_769_0,
     author = {Chang-Yu Guo and Chang-Lin Xiang},
     title = {Conservation law of harmonic mappings in supercritical dimensions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {769--773},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.592},
     language = {en},
}
TY  - JOUR
AU  - Chang-Yu Guo
AU  - Chang-Lin Xiang
TI  - Conservation law of harmonic mappings in supercritical dimensions
JO  - Comptes Rendus. Mathématique
PY  - 2024
SP  - 769
EP  - 773
VL  - 362
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.592
LA  - en
ID  - CRMATH_2024__362_G7_769_0
ER  - 
%0 Journal Article
%A Chang-Yu Guo
%A Chang-Lin Xiang
%T Conservation law of harmonic mappings in supercritical dimensions
%J Comptes Rendus. Mathématique
%D 2024
%P 769-773
%V 362
%I Académie des sciences, Paris
%R 10.5802/crmath.592
%G en
%F CRMATH_2024__362_G7_769_0
Chang-Yu Guo; Chang-Lin Xiang. Conservation law of harmonic mappings in supercritical dimensions. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 769-773. doi : 10.5802/crmath.592. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.592/

[1] Robert A. Adams; John J. F. Fournier Sobolev spaces, Pure and Applied Mathematics, 140, Academic Press Inc., 2003 | Zbl

[2] Y. Chen The weak solutions to the evolution problems of harmonic maps, Math. Z., Volume 201 (1989) no. 1, pp. 69-74 | DOI | Zbl

[3] C.-Y. Guo; C.-L. Xiang The conservation law approach in geometric PDEs, Surveys in Geometric Analysis, Volume 2022, Science Press, Beijing, 2023, pp. 51-63

[4] F. Hélein Harmonic maps, conservation laws and moving frames, Cambridge Tracts in Mathematics, 150, Cambridge University Press, 2002 | DOI | Zbl

[5] R. O’Neil Convolution operators and L(p,q) spaces, Duke Math. J., Volume 30 (1963), pp. 129-142 | DOI | Zbl

[6] T. Rivière Conservation laws for conformally invariant variational problems, Invent. Math., Volume 168 (2007), pp. 1-22 | DOI | Zbl

[7] T. Rivière; M. Struwe Partial regularity for harmonic maps and related problems, Commun. Pure Appl. Math., Volume 61 (2008), pp. 451-463 | DOI | Zbl

[8] Jacob Rubinstein; Peter Sternberg; Joseph B. Keller Reaction-diffusion processes and evolution to harmonic maps, SIAM J. Appl. Math., Volume 49 (1989), pp. 1722-1733 | DOI | Zbl

[9] J. Shatah Weak solutions and developments of singularities of the SU(2) σ-model, Commun. Pure Appl. Math., Volume 41 (1988), pp. 459-469 | DOI | Zbl

[10] K. K. Uhlenbeck Connections with L p bounds on curvature, Commun. Math. Phys., Volume 83 (1982), pp. 31-42 | DOI | Zbl

Cité par Sources :

Commentaires - Politique