Skip to main content

The role of conservation laws in the analysis of conformally invariant problems

  • Conference paper

Part of the book series: CRM Series ((CRMSNS,volume 13))

Abstract

These lecture notes form the cornerstone between two areas of Mathematics: calculus of variations and conformal invariance theory.

This is a preview of subscription content, log in via an institution.

Buying options

eBook
USD   24.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D. R. Adams, A note on Riesz potentials, DukeMath. J. 42 (1975), 765–778.

    MATH  Google Scholar 

  2. F. Bethuel, Un résultat de régularité pour les solutions de l’équation de surfaces à courbure moyenne prescrite, (French) “A regularity result for solutions to the equation of surfaces of prescribed mean curvature” C. R. Acad. Sci. Paris Sér. I Math. 314 (1992), 1003–1007.

    MathSciNet  MATH  Google Scholar 

  3. F. Bethuel, On the singular set of stationary harmonic maps, Manuscripta Math. 78 (1993), 417–443.

    Article  MathSciNet  MATH  Google Scholar 

  4. P. Choné, A regularity result for critical points of conformally invariant functionals, Potential Anal. 4 (1995), 269–296.

    Article  MathSciNet  MATH  Google Scholar 

  5. R. R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9) 72 (1993), 247–286.

    MathSciNet  MATH  Google Scholar 

  6. R. R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2) 103 (1976), 611–635.

    Article  MathSciNet  MATH  Google Scholar 

  7. F. Da Lio, “Fractional Harmonic Maps into Manifolds in odd Dimension n > 1”, arXiv, 2010.

    Google Scholar 

  8. F. Da Lio and T. Rivière, 3-Commutators estimates and the regularity of 1/2 harmonic maps into spheres, Analysis&PDE 4 (2011), 149–190.

    MathSciNet  MATH  Google Scholar 

  9. F. Da Lio and T. Rivière, Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to halfharmonic maps, Adv. Math. 227 (2011), 1300–1348.

    Article  MathSciNet  MATH  Google Scholar 

  10. C. Evans, Partial regularity for stationary harmonic maps into spheres, Arch. Rat. Mech. Anal. 116 (1991), 101–113.

    Article  MATH  Google Scholar 

  11. J. Frehse, A discontinuous solution of a midly nonlinear elliptic system, Math. Z. 134 (1973), 229–230.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Freire, S. Müller and M. Struwe, Weak convergence of wave maps from (1 + 2)-dimensional Minkowski space to Riemannian manifolds, Invent. Math. 130 (1997), 589–617.

    Article  MathSciNet  MATH  Google Scholar 

  13. Y. Gpe, Estimations of the best constant involving the L 2 norm in Wente’s inequality and compact H-Surfaces in Euclidian space, ESAIM: COCV 3 (1998), 263–300.

    Article  Google Scholar 

  14. M. Giaquinta, “Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems”, Annals of Mathematics Studies, 105, Princeton University Press, Princeton, NJ, 1983.

    Google Scholar 

  15. M. Grüter, Conformally invariant variational integrals and the removability of isolated singularities, Manuscripta Math. 47 (1984), 85–104.

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Grüter, Regularity of weak H-surfaces, J. Reine Angew. Math. 329 (1981), 1–15.

    MathSciNet  MATH  Google Scholar 

  17. E. Heinz, Ein Regularitätssatz für schwache Lösungen nichtlinearer elliptischer systeme, (German) Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1975, 1–13.

    MathSciNet  Google Scholar 

  18. E. Heinz, Uber die regularität schwacher Lösungen nichtlinearer elliptischer systeme, (German) “On the regularity of weak solutions of nonlinear elliptic systems” Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II (1986), 1–15.

    MathSciNet  Google Scholar 

  19. F. Hélein, “Harmonic Maps, Conservation Laws and Moving Frames”, Cambridge Tracts in Math. 150, Cambridge Univerity Press, 2002.

    Google Scholar 

  20. S. Hildebrandt, “Nonlinear Elliptic Systems and Harmonic Mappings”, Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, vol. 1, 2, 3, Beijing, 1980, 481–615, Science Press, Beijing, 1982.

    Google Scholar 

  21. S. Hildebrandt, “Quasilinear elliptic systems in diagonal form”, Systems of nonlinear partial differential equations (Oxford, 1982), 173–217, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 111, Reidel, Dordrecht, 1983.

    Google Scholar 

  22. T. Lamm and T. Rivière, Conservation laws for fourth order systems in four dimensions, Comm. P.D.E. 33 (2008), 245–262.

    Article  MATH  Google Scholar 

  23. T. Müller, Higher integrability of determinants and weak convergence in Literatur 1, J. Reine Angew. Math. 412 (1990), 20–34.

    MathSciNet  MATH  Google Scholar 

  24. T. Rivière, Conservation laws for conformally invariant variational problems, Invent. Math. 168 (2007), 1–22.

    Article  MathSciNet  MATH  Google Scholar 

  25. T. Rivière, Analysis aspects of Willmore surfaces, Inventiones Math. 174 (2008), 1–45.

    Article  MATH  Google Scholar 

  26. T. Rivière, Sub-criticality of Schrödinger systems with antisymmetric potentials, J. Math. Pures Appl. (9) 95 (2011), 260–276.

    MathSciNet  MATH  Google Scholar 

  27. T. Rivière and M. Struwe, Partial regularity for harmonic maps and related problems, Comm. Pure Appl. Math. 61 (2008), 451–463.

    Article  MathSciNet  MATH  Google Scholar 

  28. J. Shatah, Weak solutions and development of singularities of the SU(2) s-model, Comm. Pure Appl. Math. 41 (1988), 459–469.

    Article  MathSciNet  MATH  Google Scholar 

  29. J. Shatah and M. Struwe, The Cauchy problem for wave maps, Int. Math. Res. Not. (2002), 555–571.

    Google Scholar 

  30. T. Tao, Global regularity of wave maps. I. Small critical Sobolev norm in high dimension, Internat. Math. Res. Notices (2001), 299–328.

    Google Scholar 

  31. T. Tao, Global regularity of wave maps. II. Small energy in two dimensions, Comm. Math. Phys. 224 (2001), 443–544.

    Article  MathSciNet  MATH  Google Scholar 

  32. L. Tartar, “Remarks on Oscillations and Stokes’ Equation. Macroscopic Modelling of Turbulent Flows”, Nice, 1984, 24–31, Lecture Notes in Phys., 230, Springer, Berlin, 1985.

    Google Scholar 

  33. L. Tartar, “An Introduction to Sobolev Spaces and Interpolation Spaces”, Lecture Notes of the Unione Matematica Italiana 3, Springer, Berlin; UMI, Bologna, 2007.

    Google Scholar 

  34. P. Topping, The optimal constant in Wente’s L∞ estimate, Comm. Math. Helv. 72 (1997), 316–328.

    Article  MathSciNet  MATH  Google Scholar 

  35. K. K. Uhlenbeck, Connections with L p bounds on curvature, Comm. Math. Phys. 83 (1982), 31–42.

    Article  MathSciNet  MATH  Google Scholar 

  36. H. C. Wente, An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl. 26 (1969), 318–344.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Giuseppe Mingione

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Scuola Normale Superiore Pisa

About this paper

Cite this paper

Rivière, T. (2012). The role of conservation laws in the analysis of conformally invariant problems. In: Mingione, G. (eds) Topics in Modern Regularity Theory. CRM Series, vol 13. Edizioni della Normale. https://doi.org/10.1007/978-88-7642-427-4_2

Download citation

Publish with us

Policies and ethics