A remark on gauge transformations and the moving frame method
Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 2, p. 503-515
In this note we give a shorter proof of recent regularity results on elliptic partial differential equations with antisymmetric structure presented in Rivière (2007) [23], Rivière and Struwe (2008) [24]. We differ from the mentioned articles in using the direct method of Hélein's moving frame, i.e. minimizing a certain variational energy-functional, in order to construct a suitable gauge transformation. Though this is neither new nor surprising, it enables us to describe a proof of regularity using elementary arguments of calculus of variations and algebraic identities.Moreover, we remark that in order to prove Hildebrandt's conjecture on regularity of critical points of 2D-conformally invariant variational problems one can avoid the application of the Nash–Moser imbedding theorem.
La note contient une démonstation plus courte des résultats récents sur la régularité des solutions d'équations aux dérivées partielles ayant une structure antisymétrique comme dans Rivière (2007) [23], Rivière and Struwe (2008) [24]. La différence par rapport méthodes dans ces articles est qu'on utilise directement les « répères mobiles » developpés par Hélein, c'est – à – dire la minimisation d'une énergie variationnelle, dans la but de construire une transformation de Jauge. Même si ce n'est ni nouveau ni étonnant, ceci nous permet de mener une démonstration de régularité par des arguments élémentaires du calcul variationnel et des identités algébriques.De plus, nous remarquons que la conjecture d'Hildebrandt, concernant la regularité des points critiques des problèmes variationnels invariants sous des transformations conformes, ne nécessite pas l'application du théorème dimmersion de Nash et Moser.
DOI : https://doi.org/10.1016/j.anihpc.2009.09.004
Classification:  35J45,  35B65,  53A10
Keywords: Regularity, Systems with skew-symmetric structure, Non-linear decomposition, Moving frame
@article{AIHPC_2010__27_2_503_0,
     author = {Schikorra, Armin},
     title = {A remark on gauge transformations and the moving frame method},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {27},
     number = {2},
     year = {2010},
     pages = {503-515},
     doi = {10.1016/j.anihpc.2009.09.004},
     zbl = {1187.35054},
     mrnumber = {2595189},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2010__27_2_503_0}
}
Schikorra, Armin. A remark on gauge transformations and the moving frame method. Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 2, pp. 503-515. doi : 10.1016/j.anihpc.2009.09.004. http://www.numdam.org/item/AIHPC_2010__27_2_503_0/

[1] H. Brezis, J.-M. Coron, Multiple solutions of H-systems and Rellich's conjecture, Comm. Pure Appl. Math. 37 no. 2 (1984), 149-187 | MR 733715 | Zbl 0537.49022

[2] S. Chanillo, Sobolev inequalities involving divergence free maps, Comm. Partial Differential Equations 16 no. 12 (1991), 1969-1994 | MR 1140780 | Zbl 0778.42011

[3] P. Chone, A regularity result for critical points of conformally invariant functionals, Potential Anal. 4 (1995), 269-296 | MR 1331835 | Zbl 0833.53010

[4] S. Chanillo, Y.Y. Li, Continuity of solutions of uniformly elliptic equations in 𝐑 2 , Manuscripta Math. 77 no. 4 (1992), 415-433 | MR 1190215 | Zbl 0797.35031

[5] R. Coifman, P.-L. Lions, Y. Meyer, S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9) 72 no. 3 (1993), 247-286 | MR 1225511 | Zbl 0864.42009

[6] S. Fröhlich, F. Müller, On the existence of normal coulomb frames for two-dimensional immersions with higher codimension, preprint, 2009 | MR 2822307

[7] C. Fefferman, E.M. Stein, H p spaces of several variables, Acta Math. 129 no. 3–4 (1972), 137-193 | MR 447953 | Zbl 0257.46078

[8] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Ann. of Math. Stud. vol. 105, Princeton Univ. Press, Princeton, NJ (1983) | MR 717034 | Zbl 0516.49003

[9] M. Grüter, Conformally invariant variational integrals and the removability of isolated singularities, Manuscripta Math. 47 no. 1–3 (1984), 85-104 | MR 744314 | Zbl 0543.49020

[10] M. Günther, Isometric embeddings of Riemannian manifolds, Proceedings of the International Congress of Mathematicians, vols. I, II, Kyoto, 1990, Math. Soc. Japan, Tokyo (1991), 1137-1143 | MR 1159298 | Zbl 0745.53031

[11] R.S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 no. 1 (1982), 65-222 | MR 656198 | Zbl 0499.58003

[12] F. Hélein, Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne, C. R. Acad. Sci. Paris Sér. I 312 (1991), 591-596 | MR 1101039 | Zbl 0728.35015

[13] F. Hélein, Harmonic Maps, Conservation Laws and Moving Frames, Cambridge Tracts in Math. vol. 150, Cambridge Univ. Press (2002) | MR 1913803 | Zbl 1010.58010

[14] S. Hildebrandt, Nonlinear elliptic systems and harmonic mappings, Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, vols. 1, 2, 3, Science Press, Beijing (1982), 481-615 | MR 714341

[15] S. Hildebrandt, Quasilinear elliptic systems in diagonal form, Systems of Nonlinear Partial Differential Equations, Oxford, 1982, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. vol. 111, Reidel, Dordrecht (1983), 173-217 | MR 725522 | Zbl 0533.35026

[16] T. Iwaniec, G. Martin, Geometric Function Theory and Non-Linear Analysis, Oxford Univ. Press, Clarendon (2001) | MR 1859913

[17] N.H. Kuiper, On C 1 -isometric imbeddings. I, II, Nederl. Akad. Wetensch. Proc. Ser. A. 58 (1955), 545-556, Indag. Math. 17 (1955), 683-689 | MR 75640 | Zbl 0067.39601

[18] C.B. Morrey, Multiple Integrals in the Calculus of Variations, Grundlehren Math. Wiss. vol. 130, Springer-Verlag, Berlin (1966) | MR 202511 | Zbl 0142.38701

[19] Y. Meyer, T. Rivière, A partial regularity result for a class of stationary Yang–Mills fields in high dimension, Rev. Mat. Iberoamericana 19 no. 1 (2003), 195-219 | MR 1993420 | Zbl 1127.35317

[20] F. Müller, A. Schikorra, Boundary regularity via Uhlenbeck–Rivière decomposition, Analysis 29 (2009), 199-220 | MR 2554638 | Zbl 1181.35102

[21] S. Müller, Higher integrability of determinants and weak convergence in L 1 , J. Reine Angew. Math. 412 (1990), 20-34 | MR 1078998 | Zbl 0713.49004

[22] J. Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. (2) 63 (1956), 20-63 | MR 75639 | Zbl 0070.38603

[23] T. Rivière, Conservation laws for conformally invariant variational problems, Invent. Math. 168 no. 1 (2007), 1-22 | MR 2285745 | Zbl 1128.58010

[24] T. Rivière, M. Struwe, Partial regularity for harmonic maps and related problems, Comm. Pure Appl. Math. 61 no. 4 (2008), 451-463 | MR 2383929 | Zbl 1144.58011

[25] S. Semmes, A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller, Comm. Partial Differential Equations 19 no. 1–2 (1994), 277-319 | MR 1257006 | Zbl 0836.35030

[26] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. With the assistance of Timothy S. Murphy, Princeton Math. Ser. vol. 43, Princeton Univ. Press, Princeton, NJ (1993) | MR 1232192 | Zbl 0821.42001

[27] L. Tartar, Remarks on oscillations and Stokes' equation, Macroscopic Modelling of Turbulent Flows, Proceedings, Lecture Notes in Phys. vol. 230, Sophia-Antipolis, France (1984), 24-31 | MR 815930

[28] K.K. Uhlenbeck, Connections with L p bounds on curvature, Comm. Math. Phys. 83 no. 1 (1982), 31-42 | MR 648356 | Zbl 0499.58019

[29] C. Wang, A compactness theorem of n-harmonic maps, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 no. 4 (2005), 509-519 | MR 2145723 | Zbl 1229.58017

[30] H.C. Wente, An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl. 26 (1969), 318-344 | MR 243467 | Zbl 0181.11501