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Very weak solutions to the two-dimensional Monge-Ampére equation

  • Wentao Cao
  • László SzékelyhidiJr.Email author
Articles
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Abstract

In this short note we revisit the convex integration approach to constructing very weak solutions to the 2D Monge-Ampére equation with Hölder-continuous first derivatives of exponent β < 1/5. Our approach is based on combining the approach of Lewicka and Pakzad (2017) with a new diagonalization procedure which avoids the use of conformai coordinates, which was introduced by De Lellis et al. (2018) for the isometric immersion problem.

Keywords

Monge-Ampére equation convex integration weak solutions 

MSC(2010)

35M10 76B03 76F02 

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Notes

Acknowledgements

The authors thank the hospitality of the Max-Planck Institute for Mathematics in the Sciences, and gratefully acknowledge the support of the European Research Council Grant Agreement (Grant No. 724298).

@@@References

  1. 1.
    Ambrosio L. Well posedness of ODE’s and continuity equations with nonsmooth vector fields, and applications. Rev Mat Complut, 2017, 30: 427–450MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Borisov Y F. C1, α-isometric immersions of Riemannian spaces. Dokl Akad Nauk SSSR (NS), 1965, 163: 11–13Google Scholar
  3. 3.
    Buckmaster T, De Lellis C, Székelyhidi Jr L, et al. Onsager’s conjecture for admissible weak solutions. Comm Pure Appl Math, 2019, in pressGoogle Scholar
  4. 4.
    Buckmaster T, Shkoller S, Vicol V. Nonuniqueness of weak solutions to the SQG equation. Comm Pure Appl Math, 2019, in pressGoogle Scholar
  5. 5.
    Cao W, Székelyhidi Jr L. C1, α isometric extensions. Comm Partial Differential Equations, 2019, in pressGoogle Scholar
  6. 6.
    Constantin P E W, Titi E S. Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Comm Math Phys, 1994, 165: 207–209MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Conti S, De Lellis C, Székelyhidi Jr L. h-principle and rigidity for C 1, α isometric embeddings. In: Nonlinear Partial Differential Equations. Abel Symposia, vol. 7. Berlin-Heidelberg: Springer, 2012, 83–116CrossRefGoogle Scholar
  8. 8.
    De Lellis C, Inauen D, Székelyhidi Jr L. A Nash-Kuiper theorem for C 1, 1/5–δ immersions of surfaces in 3 dimensions. ArXiv:1510.01934v2, 2016Google Scholar
  9. 9.
    Depauw N. Non unicité des solutions bornées pour un champ de vecteurs BV en dehors d’un hyperplan. C R Math Acad Sci Paris, 2003, 337: 249–252MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    DiPerna R J, Lions P-L. Ordinary differential equations, transport theory and Sobolev spaces. Invent Math, 1989, 98: 511–547MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Han Q, Hong J-X. Isometric Embedding of Riemannian Manifolds in Euclidean Spaces. Mathematical Surveys and Monographs, vol. 130. Providence: Amer Math Soc, 2006Google Scholar
  12. 12.
    Herglotz G. Über die starrheit der eiflächen. Abh Math Sem Univ Hamburg, 1943, 15: 127–129MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hungerbuhler N, Wasem M. The one-sided isometric extension problem. Results Math, 2017, 71: 749–781MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Isett P. A proof of Onsager’s conjecture. Ann of Math (2), 2018, 188: 1–93MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Iwaniec T. On the concept of the weak Jacobian and Hessian. Report Univ Jyväskylä, 2001, 83: 181–205MathSciNetzbMATHGoogle Scholar
  16. 16.
    Jerrard R L. Some rigidity results related to Monge-Ampère functions. Canad J Math, 2009, 62: 320–354CrossRefzbMATHGoogle Scholar
  17. 17.
    Kirchheim B. Rigidity and geometry of microstructures. Habilitation Thesis. Leipzig: University of Leipzig, 2003Google Scholar
  18. 18.
    Kuiper N H. On C 1-isometric imbeddings. I. Nederl Akad Wetensch Indag Math, 1955, 17: 545–556CrossRefzbMATHGoogle Scholar
  19. 19.
    Kuiper N H. On C 1-isometric imbeddings. II. Nederl Akad Wetensch Indag Math, 1955, 17: 683–689CrossRefzbMATHGoogle Scholar
  20. 20.
    Lewicka M, Pakzad M R. Convex integration for the Monge-Ampère equation in two dimensions. Anal PDE, 2017, 3: 695–727CrossRefzbMATHGoogle Scholar
  21. 21.
    Lions J-L, Temam R, Wang S H. On the equations of the large-scale ocean. Nonlinearity, 1992, 5: 1007–1053MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Modena S, Székelyhidi Jr L. Non-uniqueness for the transport equation with Sobolev vector fields. Ann PDE, 2019, in pressGoogle Scholar
  23. 23.
    Nash J. C 1 isometric imbeddings. Ann of Math (2), 1954, 60: 383–396MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Pakzad M R. On the Sobolev space of isometric immersions. J Differential Geom, 2004, 66: 47–69MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Resnick S G. Dynamical problems in non-linear advective partial differential equations. PhD Thesis. Chicago: University of Chicago, 1995Google Scholar
  26. 26.
    Šverák V. On regularity for the Monge-Ampère equation without convexity assumptions. Technical Report. Edinburgh: Heriot-Watt University, 1991Google Scholar
  27. 27.
    Székelyhidi Jr. L. From isometric embeddings to turbulence. In: HCDTE Lecture Notes. Part II. Nonlinear Hyperbolic PDEs, Dispersive and Transport Equations. Springfield: Am Inst Math Sci, 2014, 1–66Google Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut für MathematikUniversität LeipzigLeipzigGermany

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