A global weak solution of the Dirac-harmonic map flow

Jost, Jürgen; Liu, Lei; Zhu, Miaomiao

doi:10.1016/j.anihpc.2017.01.002

Jost, Jürgen ; Liu, Lei ; Zhu, Miaomiao

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 7, pp. 1851-1882.

Résumé

We show the existence of a global weak solution of the heat flow for Dirac-harmonic maps from compact Riemann surfaces with boundary when the energy of the initial map and the $L^{2}$ -norm of the boundary values of the spinor are sufficiently small. Dirac-harmonic maps couple a second order harmonic map type system with a first-order Dirac type system. The heat flow that has been introduced in [9] and that we investigate here is novel insofar as we only make the second order part parabolic, but carry the first order part along the resulting flow as an elliptic constraint. Of course, since the equations are coupled, both parts then change along the flow.

The solution is unique and regular with the exception of at most finitely many singular times. We also discuss the behavior at the singularities of the flow.

As an application, we deduce some existence results for Dirac-harmonic maps. Since we may impose nontrivial boundary conditions also for the spinor part, in the limit, we shall obtain Dirac-harmonic maps with nontrivial spinor part.

MR Zbl

DOI : 10.1016/j.anihpc.2017.01.002

Classification : 53C43, 58E20
Mots clés : Dirac-harmonic map, Dirac-harmonic flow, Blow-up, Dirichlet boundary, Chiral boundary, Singularity

@article{AIHPC_2017__34_7_1851_0,
     author = {Jost, J\"urgen and Liu, Lei and Zhu, Miaomiao},
     title = {A global weak solution of the {Dirac-harmonic} map flow},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1851--1882},
     publisher = {Elsevier},
     volume = {34},
     number = {7},
     year = {2017},
     doi = {10.1016/j.anihpc.2017.01.002},
     mrnumber = {3724759},
     zbl = {1380.53068},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2017.01.002/}
}

TY  - JOUR
AU  - Jost, Jürgen
AU  - Liu, Lei
AU  - Zhu, Miaomiao
TI  - A global weak solution of the Dirac-harmonic map flow
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2017
SP  - 1851
EP  - 1882
VL  - 34
IS  - 7
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2017.01.002/
DO  - 10.1016/j.anihpc.2017.01.002
LA  - en
ID  - AIHPC_2017__34_7_1851_0
ER  -

%0 Journal Article
%A Jost, Jürgen
%A Liu, Lei
%A Zhu, Miaomiao
%T A global weak solution of the Dirac-harmonic map flow
%J Annales de l'I.H.P. Analyse non linéaire
%D 2017
%P 1851-1882
%V 34
%N 7
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2017.01.002/
%R 10.1016/j.anihpc.2017.01.002
%G en
%F AIHPC_2017__34_7_1851_0

Jost, Jürgen; Liu, Lei; Zhu, Miaomiao. A global weak solution of the Dirac-harmonic map flow. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 7, pp. 1851-1882. doi : 10.1016/j.anihpc.2017.01.002. http://www.numdam.org/articles/10.1016/j.anihpc.2017.01.002/

Bibliographie
Cité par

[1] Adams, R.A. Sobolev Spaces, Pure and Applied Mathematics, vol. 65, Academic Press, 1975 | MR | Zbl

[2] Ammann, B.; Ginoux, N. Dirac-harmonic maps from index theory, Calc. Var. Partial Differ. Equ., Volume 47 (2013) no. 3–4, pp. 739–762 | MR | Zbl

[3] Branding, V. The evolution equations for regularized Dirac-geodesics, J. Geom. Phys., Volume 100 (2016), pp. 1–19 | DOI | MR | Zbl

[4] Branding, V. On the evolution of regularized Dirac-harmonic maps from closed surfaces | arXiv | DOI | Zbl

[5] Chang, K.C. Heat flow and boundary value problem for harmonic maps, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 6 (1989) no. 5, pp. 363–395 | Numdam | MR | Zbl

[6] Chen, Q.; Jost, J.; Wang, G. The maximum principle and the Dirichlet problem for Dirac-harmonic maps, Calc. Var. Partial Differ. Equ., Volume 47 (2013) no. 12, pp. 87–116 | MR | Zbl

[7] Chen, Q.; Jost, J.; Li, J.; Wang, G. Regularity theorems and energy identities for Dirac-harmonic maps, Math. Z., Volume 251 (2005) no. 1, pp. 61–84 | DOI | MR | Zbl

[8] Chen, Q.; Jost, J.; Li, J.; Wang, G. Dirac-harmonic maps, Math. Z., Volume 254 (2006) no. 2, pp. 409–432 | DOI | MR | Zbl

[9] Q. Chen, J. Jost, L. Sun, M. Zhu, Estimates for solutions of Dirac equations and an application to a geometric elliptic-parabolic problem, MPI MIS Preprint: 79/2014. | MR

[10] Chen, Q.; Jost, J.; Sun, L.; Zhu, M. Dirac-geodesics and their heat flows, Calc. Var. Partial Differ. Equ., Volume 54 (2015) no. 3, pp. 2615–2635 | DOI | MR | Zbl

[11] Chen, Q.; Jost, J.; Wang, G.; Zhu, M. The boundary value problem for Dirac-harmonic maps, J. Eur. Math. Soc., Volume 15 (2013) no. 3, pp. 997–1031 | DOI | MR | Zbl

[12] Chen, Y.; Levine, S. The existence of the heat flow of H-systems, Discrete Contin. Dyn. Syst., Volume 8 (2002) no. 1, pp. 219–236 | DOI | MR | Zbl

[13] Gastel, A. The extrinsic polyharmonic map heat flow in the critical dimension, Adv. Geom., Volume 6 (2006) no. 4, pp. 501–521 | DOI | MR | Zbl

[14] Hamilton, R.S. Harmonic Maps of Manifolds with Boundary, Lecture Notes in Mathematics, vol. 471, Springer-Verlag, Berlin–New York, 1975 | DOI | MR | Zbl

[15] Jost, J.; Mo, X.; Zhu, M. Some explicit constructions of Dirac-harmonic maps, J. Geom. Phys., Volume 59 (2009) no. 11, pp. 1512–1527 | DOI | MR | Zbl

[16] J. Jost, L. Liu, M. Zhu, Geometric analysis of the action functional of the nonlinear supersymmetric sigma model, MPI MIS Preprint: 77/2015.

[17] Ladyzenskaja, O.; Solonnikov, V.A.; Ural'ceva, N.N. Linear and Quasilinear Equations of Parabolic Type, AMS Transl. Math. Monogr., vol. 23, AMS, Providence, RI, 1968 | MR | Zbl

[18] Lawson, H.B.; Michelsohn, M.L. Spin Geometry, vol. 38, Princeton University Press, 1989 | MR | Zbl

[19] Lemaire, L. Applications harmoniques de surfaces riemanniennes, J. Differ. Geom., Volume 13 (1978), pp. 51–78 | DOI | MR | Zbl

[20] Lin, F.; Wang, C. The Analysis of Harmonic Maps and Their Heat Flows, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008 (xii+267 pp.) | DOI | MR | Zbl

[21] Müller, F.; Schikorra, A. Boundary regularity via Uhlenbeck–Rivière decomposition, Analysis, Volume 29 (2009) no. 2, pp. 199–220 | DOI | MR | Zbl

[22] Schoen, R. Seminar on Nonlinear Partial Differential Equations, Math. Sci. Res. Inst. Publ., Volume vol. 2, Springer, New York (1984), pp. 321–358 (Berkeley, Calif., 1983) | MR | Zbl

[23] Sharp, B.; Zhu, M. Regularity at the free boundary for Dirac-harmonic maps from surfaces, Calc. Var. Partial Differ. Equ., Volume 55 (2016) no. 2, pp. 27 | DOI | MR | Zbl

[24] Struwe, M. On the evolution of harmonic mappings of Riemannian surfaces, Commun. Math. Helv., Volume 60 (1985), pp. 558–581 | DOI | MR | Zbl

[25] Struwe, M. The existence of surfaces of constant mean curvature with free boundaries, Acta Math., Volume 160 (1988) no. 1–2, pp. 19–64 | MR | Zbl

[26] Wang, C. Heat flow of biharmonic maps in dimension four and its application, Pure Appl. Math. Q., Volume 3 (2007) no. 2, pp. 595–613 | DOI | MR | Zbl

[27] Wang, C.; Xu, D. Regularity of Dirac-harmonic maps, Int. Math. Res. Not. (2009) no. 20, pp. 3759–3792 | MR | Zbl

[28] Zhu, M. Regularity of weakly Dirac-harmonic maps to hypersurfaces, Ann. Glob. Anal. Geom., Volume 35 (2009) no. 4, pp. 405–412 | MR | Zbl

Cité par Sources :