Skip to main content
SpringerLink
Log in

Some Singular Minimizers in Low Dimensions in the Calculus of Variations

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We construct a singular minimizing map u from \({\mathbb{R}^{3}}\) to \({\mathbb{R}^{2}}\) of a smooth uniformly convex functional of the form \({\int_{B_1} F(D{\bf u})\,dx}\).

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ambrosio L., Soner H.M.: Level set approach to mean curvature flow in arbitrary codimension. J. Differ. Geom. 43, 693–737 (1996)

    MathSciNet  MATH  Google Scholar 

  2. Colombo M., Figalli A.: Regularity results for very degenerate elliptic equations. J. Math. Pures Appl. (9) 101(1), 94–117 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. De Giorgi E.: Sulla differenziabilità e l’analicità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino Cl. Sci. Fis. Fat. Nat. 3, 25–43 (1957)

    MathSciNet  MATH  Google Scholar 

  4. De Giorgi E.: Un esempio di estremali discontinue per un problema variazionale di tipo ellittico. Boll. UMI 4, 135–137 (1968)

    MathSciNet  MATH  Google Scholar 

  5. De Silva D., Savin O.: Minimizers of convex functionals arising in random surfaces. Duke Math. J. 151(3), 487–532 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Evans L.C.: A new proof of local \({C^{1,\alpha}}\) regularity for solutions of certain degenerate elliptic P.D.E. J. Differ. Equ. 45(3), 356–373 (1982)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Evans L.C.: Quasiconvexity and partial regularity in the calculus of variations. Arch. Rational Mech. Anal. 95(3), 227–252 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Giaquinta M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press, Princeton (1983)

    MATH  Google Scholar 

  9. Giaquinta M., Giusti E.: Nonlinear elliptic systems with quadratic growth. Manuscr. Math. 24, 323–349 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  10. Giusti E., Miranda M.: Sulla regolarità delle soluzioni deboli di una classe di sistemi ellittici quasi-lineari. Arch. Rational Mech. Anal. 31, 173–184 (1968)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Giusti E., Miranda M.: Un esempio di soluzione discontinua per un problem di minimo relativo ad un integrale regolare del calcolo delle variazioni. Boll. Un. Mat. Ital. 2, 1–8 (1968)

    MathSciNet  MATH  Google Scholar 

  12. Hao W., Leonardi S., Necas J.: An example of an irregular solution to a nonlinear Euler–Lagrange elliptic system with real analytic coefficients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23(1), 57–67 (1996)

    MathSciNet  MATH  Google Scholar 

  13. Morrey C.B.: Multiple Integrals in the Calculus of Variations. Springer, Heidelberg (1966)

    MATH  Google Scholar 

  14. Nash J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  15. Necas, J.: Example of an irregular solution to a nonlinear elliptic system with analytic coefficients and conditions of regularity. In: Theory of Non Linear Operators, Abhandlungen Akad. der Wissen. der DDR (1997) (proceedings of a summer school held in Berlin, 1975)

  16. Šverák V., Yan X.: A singular minimizer of a smooth strongly convex functional in three dimensions. Cal. Var. PDE 10(3), 213–221 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Šverák V., Yan X.: Non-Lipschitz minimizers of smooth uniformly convex functionals. Proc. Natl. Acad. Sci. USA 99(24), 15269–15276 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, UT Austin, Austin, TX, 78712, USA

    Connor Mooney

  2. Department of Mathematics, Columbia University, New York, NY, 10027, USA

    Ovidiu Savin

Authors
  1. Connor Mooney
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ovidiu Savin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Connor Mooney.

Additional information

Communicated by V. Šverák

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mooney, C., Savin, O. Some Singular Minimizers in Low Dimensions in the Calculus of Variations. Arch Rational Mech Anal 221, 1–22 (2016). https://doi.org/10.1007/s00205-015-0955-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-015-0955-x

Keywords

Search

Navigation