Analyse des problèmes conformément invariants
Séminaire Laurent Schwartz — EDP et applications (2016-2017), Exposé no. 12, 26 p.

Cet exposé constitue une revue d’une technique développée avec T. Rivière pour prouver des identités d’énergie pour les limites de suites de solutions de problèmes conformément invariants. Le point de départ est [34] où l’on prouve de telles identités pour tous les problèmes conformément invariants en dimension 2. Contrairement aux résultats existants, la preuve repose exclusivement sur l’invariance conforme. Elle a pu être transposée à beaucoup de problèmes ouverts en dimension supérieure, d’ordre supérieur ou encore à bord libre.

Publié le :
DOI : https://doi.org/10.5802/slsedp.110
@article{SLSEDP_2016-2017____A12_0,
     author = {Laurain, Paul},
     title = {Analyse des probl\`emes conform\'ement invariants},
     journal = {S\'eminaire Laurent Schwartz --- EDP et applications},
     note = {talk:12},
     publisher = {Institut des hautes \'etudes scientifiques \& Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2016-2017},
     doi = {10.5802/slsedp.110},
     language = {fr},
     url = {http://www.numdam.org/item/SLSEDP_2016-2017____A12_0/}
}
Laurain, Paul. Analyse des problèmes conformément invariants. Séminaire Laurent Schwartz — EDP et applications (2016-2017), Exposé no. 12, 26 p. doi : 10.5802/slsedp.110. http://www.numdam.org/item/SLSEDP_2016-2017____A12_0/

[2] Yann Bernard, Noether’s theorem and the Willmore functional, Adv. Calc. Var. 9 (2016), no. 3, 217–234. | Article | MR 3518329 | Zbl 1343.58008

[3] Yann Bernard and Tristan Rivière, Energy quantization for Willmore surfaces and applications, Ann. of Math. (2) 180 (2014), no. 1, 87–136. | Article | MR 3194812 | Zbl 1325.53014

[4] Wilhelm Blaschke, Topologische Fragen der Differentialgeometrie X. Kurvenscharen im Raum, Abh. Math. Sem. Univ. Hamburg 7 (1929), no. 1, 37–45. | Article | MR 3069514 | Zbl 55.1024.03

[5] H. Brezis and J.-M. Coron, Convergence of solutions of H-systems or how to blow bubbles, Arch. Rational Mech. Anal. 89 (1985), no. 1, 21–56. | Article | MR 784102 | Zbl 0584.49024

[6] Haïm Brezis and Jean-Michel Coron, Multiple solutions of H-systems and Rellich’s conjecture, Comm. Pure Appl. Math. 37 (1984), no. 2, 149–187. | Article | Zbl 0537.49022

[7] Peter Buser, Geometry and spectra of compact Riemann surfaces, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2010, Reprint of the 1992 edition. | Zbl 1239.32001

[8] Paolo Caldiroli and Roberta Musina, H-bubbles in a perturbative setting : the finite-dimensional reduction method, Duke Math. J. 122 (2004), no. 3, 457–484. | Article | MR 2057016 | Zbl 1079.53012

[9] Qun Chen, Jürgen Jost, Guofang Wang, and Miaomiao Zhu, The boundary value problem for Dirac-harmonic maps, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 3, 997–1031. | Article | MR 3085099 | Zbl 1271.81125

[10] S. S. Chern and Wen Tsün Wu (eds.), Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations. Vol. 1, 2, 3, Science Press, Beijing ; Gordon & Breach Science Publishers, New York, 1982, Held in Beijing, August 18–September 21, 1980. | Article

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

[12] Tobias H. Colding and William P. Minicozzi, II, Width and finite extinction time of Ricci flow, Geom. Topol. 12 (2008), no. 5, 2537–2586. | Article | MR 2460871 | Zbl 1161.53352

[13] —, Width and finite extinction time of Ricci flow, Geom. Topol. 12 (2008), no. 5, 2537–2586. | Article | MR 2460871 | Zbl 1161.53352

[14] Francesca Da Lio, Compactness and bubble analysis for 1/2-harmonic maps, Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), no. 1, 201–224. | Article | Numdam | MR 3303947 | Zbl 1310.58011

[15] Francesca Da Lio, Paul Laurain, and Tristan Rivière, A Pohozaev-type formula and quantization of horizontal half-harmonic maps, arXiv :1607.05504 (2016).

[16] Francesca Da Lio and Tristan Rivière, Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps, Adv. Math. 227 (2011), no. 3, 1300–1348. | Article | Zbl 1219.58004

[17] Ailana Fraser and Richard Schoen, Minimal surfaces and eigenvalue problems, Geometric analysis, mathematical relativity, and nonlinear partial differential equations, Contemp. Math., vol. 599, Amer. Math. Soc., Providence, RI, 2013, pp. 105–121. | Article | Zbl 1321.35118

[18] Mariano Giaquinta and Luca Martinazzi, An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, second ed., Appunti. Scuola Normale Superiore di Pisa (Nuova Serie), vol. 11, Edizioni della Normale, Pisa, 2012. | Article | Zbl 1262.35001

[19] Loukas Grafakos, Classical Fourier analysis, third ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2014. | Article | Zbl 1304.42001

[20] M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347. | Article | MR 809718 | Zbl 0592.53025

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

[22] Frédéric Hélein, Constant mean curvature surfaces, harmonic maps and integrable systems, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001, Notes taken by Roger Moser. | Article | Zbl 1158.53301

[23] —, Harmonic maps, conservation laws and moving frames, second ed., Cambridge Tracts in Mathematics, vol. 150, Cambridge University Press, Cambridge, 2002, Translated from the 1996 French original, With a foreword by James Eells. | Article

[24] W. Helfrich, Elastic properties of lipid bilayers : theory and possible experiments, Zeitschrift für Naturforschung. Teil C : Biochemie, Biophysik, Biologie, Virologie 28(11), 693 (1973). | Article

[25] Jay Hineman, Tao Huang, and Chang-You Wang, Regularity and uniqueness of a class of biharmonic map heat flows, Calc. Var. Partial Differential Equations 50 (2014), no. 3-4, 491–524. | Article | MR 3216822 | Zbl 1298.35053

[26] Peter Hornung and Roger Moser, Energy identity for intrinsically biharmonic maps in four dimensions, Anal. PDE 5 (2012), no. 1, 61–80. | Article | MR 2957551 | Zbl 1273.58007

[27] Jürgen Jost, Lei Liu, and Miaomiao Zhu, The qualitative behavior at the free boundary for approximate harmonic maps from surfaces, MIS-Preprint 26/2016 (2016). | Article | MR 3961307 | Zbl 1425.53077

[28] Yvette Kosmann-Schwarzbach, The Noether theorems, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, New York, 2011, Invariance and conservation laws in the twentieth century, Translated, revised and augmented from the 2006 French edition by Bertram E. Schwarzbach. | Article | Zbl 1216.01011

[29] Tobias Lamm and Longzhi Lin, Estimates for the energy density of critical points of a class of conformally invariant variational problems, Adv. Calc. Var. 6 (2013), no. 4, 391–413. | Article | MR 3199733 | Zbl 1283.35031

[30] Paul Laurain and Longzhi Lin, Convexity of the biharmonic functionals and applications, In preparation (2017).

[31] Paul Laurain and Romain Petrides, Regularity and quantification for harmonic maps with free boundary, Accepté à Advances in Calculus of Variations (2015). | Article | MR 3592578 | Zbl 1370.58005

[32] —, Convexity of the energy around free-boundary harmonic maps and applications, In preparation (2017).

[33] Paul Laurain and Tristan Rivière, Energy quantization for biharmonic maps, Adv. Calc. Var. 6 (2013), no. 2, 191–216. | Article | MR 3043576 | Zbl 1275.35098

[34] —, Angular energy quantization for linear elliptic systems with antisymmetric potentials and applications, Anal. PDE 7 (2014), no. 1, 1–41. | Article | MR 3219498 | Zbl 1295.35204

[35] —, Optimal estimate for the gradient of Green functions on degenerating surfaces and applications, Accepté à CAG (2014). | Article | MR 3853930

[36] —, Energy quantization of Willmore surfaces at the boundary of the moduli space, arXiv :1606.08004, 2016. | Article | MR 3843372

[37] Fernando C. Marques and André Neves, The Willmore conjecture, Jahresber. Dtsch. Math.-Ver. 116 (2014), no. 4, 201–222. | Article | MR 3280571 | Zbl 1306.53005

[38] Thomas H. Parker, Bubble tree convergence for harmonic maps, J. Differential Geom. 44 (1996), no. 3, 595–633. | Article | MR 1431008 | Zbl 0874.58012

[39] Romain Petrides, Maximizing Steklov eigenvalues on surfaces, soumis (2015). | Article | MR 3998908 | Zbl 07104704

[40] S. I. Pohožaev, On the eigenfunctions of the equation Δu+λf(u)=0, Dokl. Akad. Nauk SSSR 165 (1965), 36–39.

[41] Tristan Rivière, Conservation laws for conformally invariant variational problems, Invent. Math. 168 (2007), no. 1, 1–22. | Article | MR 2285745 | Zbl 1128.58010

[42] —, Analysis aspects of Willmore surfaces, Invent. Math. 174 (2008), no. 1, 1–45. | Article | MR 2430975 | Zbl 1155.53031

[43] —, Conformally invariant variational problems, https://people.math.ethz.ch/~riviere/papers/conformal-course.pdf, 2012.

[44] J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math. (2) 113 (1981), no. 1, 1–24. | Article | MR 604040 | Zbl 0462.58014

[45] Armin Schikorra, A remark on gauge transformations and the moving frame method, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), no. 2, 503–515. | Article | Numdam | MR 2595189 | Zbl 1187.35054

[46] —, ε-regularity for systems involving non-local, antisymmetric operators, Calc. Var. Partial Differential Equations 54 (2015), no. 4, 3531–3570. | Article | MR 3426086 | Zbl 1348.58009

[47] Jean-Claude Sikorav, Some properties of holomorphic curves in almost complex manifolds, Holomorphic curves in symplectic geometry, Progr. Math., vol. 117, Birkhäuser, Basel, 1994, pp. 165–189. | Article

[48] Michael Struwe, Large H-surfaces via the mountain-pass-lemma, Math. Ann. 270 (1985), no. 3, 441–459. | Article | Zbl 0582.58010

[49] —, Plateau’s problem and the calculus of variations, Mathematical Notes, vol. 35, Princeton University Press, Princeton, NJ, 1988. | Zbl 0694.49028

[50] —, Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems, fourth ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 34, Springer-Verlag, Berlin, 2008.

[51] Luc Tartar, Remarks on oscillations and Stokes’ equation, Macroscopic modelling of turbulent flows (Nice, 1984), Lecture Notes in Phys., vol. 230, Springer, Berlin, 1985, pp. 24–31. | Article

[52] Karen K. Uhlenbeck, Connections with L p bounds on curvature, Comm. Math. Phys. 83 (1982), no. 1, 31–42. | Article | Zbl 0499.58019

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

[54] Brian White, Introduction to minimal surface theory, Geometric analysis, IAS/Park City Math. Ser., vol. 22, Amer. Math. Soc., Providence, RI, 2016, pp. 387–438. | Article | Zbl 1354.49091

[55] T. J. Willmore, Riemannian geometry, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993. | Zbl 0797.53002

[56] Miaomiao Zhu, Harmonic maps from degenerating Riemann surfaces, Math. Z. 264 (2010), no. 1, 63–85. | Article | MR 2564932 | Zbl 1213.53086

[57] —, Regularity for harmonic maps into certain pseudo-Riemannian manifolds, J. Math. Pures Appl. (9) 99 (2013), no. 1, 106–123. | Article | MR 3003285 | Zbl 1270.58012