Archive for Rational Mechanics and Analysis

, Volume 230, Issue 1, pp 321–342 | Cite as

Radial Symmetry of p-Harmonic Minimizers

  • Aleksis Koski
  • Jani Onninen


“It is still not known if the radial cavitating minimizers obtained by Ball (Philos Trans R Soc Lond A 306:557–611, 1982) (and subsequently by many others) are global minimizers of any physically reasonable nonlinearly elastic energy”. This quotation is from Sivaloganathan and Spector (Ann Inst Henri Poincaré Anal Non Linéaire 25(1):201–213, 2008) and seems to be still accurate. The model case of the p-harmonic energy is considered here. We prove that the planar radial minimizers are indeed the global minimizers provided we prescribe the admissible deformations on the boundary. In the traction free setting, however, even the identity map need not be a global minimizer.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adamowicz, T.: On the Geometry of p-Harmonic Mappings, Ph.D. thesis, Syracuse University, 2008Google Scholar
  2. 2.
    Antman, S.S.: Nonlinear Problems of Elasticity. Applied Mathematical Sciences, Vol. 107. Springer, New York, 1995Google Scholar
  3. 3.
    Astala, K., Iwaniec, T., Martin, G.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton University Press, Princeton (2009)zbMATHGoogle Scholar
  4. 4.
    Astala, K., Iwaniec, T., Martin, G.: Deformations of annuli with smallest mean distortion. Arch. Ration. Mech. Anal. 195(3), 899–921 (2010)CrossRefzbMATHGoogle Scholar
  5. 5.
    Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63(4), 337–403 (1976/1977)Google Scholar
  6. 6.
    Ball, J.M.: Global invertibility of Sobolev functions and the interpenetration of matter. Proc. R. Soc. Edinburgh Sect. A 88(3–4), 315–328 (1981)CrossRefzbMATHGoogle Scholar
  7. 7.
    Ball, J.M.: Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos. Trans. R. Soc. Lond. A 306, 557–611 (1982)CrossRefzbMATHGoogle Scholar
  8. 8.
    Ball, J.M.: Constitutive inequalities and existence theorems in nonlinear elastostatics. Nonlinear Analysis and Mechanics: Heriot–Watt Symposium (Edinburgh, 1976), Vol. 1. Res. Notes in Math., No. 17. Pitman, London, 187–241, 1977Google Scholar
  9. 9.
    Ball, J.M.: Existence of solutions in finite elasticity. Proceedings of the IUTAM Symposium on Finite Elasticity. Martinus Nijhoff, 1981Google Scholar
  10. 10.
    Ball, J.M.: Minimizers and the Euler–Lagrange equations. Trends and Applications of Pure Mathematics to Mechanics (Palaiseau, 1983), 1–4, Lecture Notes in Physics, Vol. 195. Springer, Berlin, 1984Google Scholar
  11. 11.
    Ball, J.M.: Some open problems in elasticity. Geometry, mechanics, and dynamics. Springer, New York, 3–59, 2002Google Scholar
  12. 12.
    Chen, Y.W.: Discontinuity and representations of minimal surface solutions. Proceedings of the Conference on Differential Equations (Dedicated to A. Weinstein). University of Maryland, College Park, 115–138, 1956Google Scholar
  13. 13.
    Ciarlet, P.G.: Mathematical elasticity Vol. I. Three-dimensional elasticity, Studies in Mathematics and its Applications, Vol. 20. North-Holland Publishing Co., Amsterdam, 1988Google Scholar
  14. 14.
    Coron, J.-M., Gulliver, R.D.: Minimizing p-harmonic maps into spheres. J. Reine Angew. Math. 401, 82–100 (1989)zbMATHGoogle Scholar
  15. 15.
    Cuneo, D.: Mappings Between Annuli of Mimimal p-Harmonic Energy. Ph.D. thesis, Syracuse University, 2017Google Scholar
  16. 16.
    Hardt, R., Lin, F.H., Wang, C.Y.: The p-energy minimality of \(x/|x|\). Commun. Anal. Geom. 6, 141–152 (1998)CrossRefzbMATHGoogle Scholar
  17. 17.
    Hencl, S., Koskela, P.: Regularity of the inverse of a planar Sobolev homeomorphism. Arch. Ration. Mech. Anal. 1 80(1), 75–95 (2006)Google Scholar
  18. 18.
    Hencl, S., Pratelli, A.: Diffeomorphic Approximation of \(W^{1,1}\) Planar Sobolev Homeomorphisms. J. Eur. Math. Soc. (to appear)Google Scholar
  19. 19.
    Hildebrandt, S., Nitsche, J.C.C.: A uniqueness theorem for surfaces of least area with partially free boundaries on obstacles. Arch. Rational Mech. Anal. 79(3), 189–218 (1982)CrossRefzbMATHGoogle Scholar
  20. 20.
    Hong, M.-C.: On the minimality of the p-harmonic map \(\frac{x}{|x|} : B^n \rightarrow S^{n-1}\). Calc. Var. Partial Differ. Equ. 13, 459–468 (2001)zbMATHGoogle Scholar
  21. 21.
    Iwaniec, T., Martin, G.: Geometric Function Theory and Non-linear Analysis. Oxford University Press, Oxford Mathematical Monographs (2001)zbMATHGoogle Scholar
  22. 22.
    Iwaniec, T., Kovalev, L.V., Onninen, J.: The Nitsche conjecture. J. Am. Math. Soc. 24(2), 345–373 (2011)CrossRefzbMATHGoogle Scholar
  23. 23.
    Iwaniec, T., Kovalev, L.V., Onninen, J.: Diffeomorphic approximation of Sobolev homeomorphisms. Arch. Ration. Mech. Anal. 201(3), 1047–1067 (2011)CrossRefzbMATHGoogle Scholar
  24. 24.
    Iwaniec, T., Kovalev, L.V., Onninen, J.: Doubly connected minimal surfaces and extremal harmonic mappings. J. Geom. Anal. 22(3), 726–762 (2012)CrossRefzbMATHGoogle Scholar
  25. 25.
    Iwaniec, T., Onninen, J.: Hyperelastic deformations of smallest total energy. Arch. Ration. Mech. Anal. 194(3), 927–986 (2009)CrossRefzbMATHGoogle Scholar
  26. 26.
    Iwaniec, T., Onninen, J.: Neohookean deformations of annuli, existence, uniqueness and radial symmetry. Math. Ann. 348(1), 35–55 (2010)CrossRefzbMATHGoogle Scholar
  27. 27.
    Iwaniec, T., Onninen, J.: An invitation to n-harmonic hyperelasticity. Pure Appl. Math. Q. 7 (2011), Special Issue: In honor of Frederick W. Gehring, Part 2, 319–343Google Scholar
  28. 28.
    Iwaniec, T., Onninen, J.: \(n\)-Harmonic mappings between annuli Mem. Am. Math. Soc. 218 (2012)Google Scholar
  29. 29.
    Iwaniec, T., Onninen, J.: Limits of Sobolev homeomorphisms. J. Eur. Math. Soc. (JEMS) 19(2), 473–505 (2017)CrossRefzbMATHGoogle Scholar
  30. 30.
    Jordens, M., Martin, G.J.: Deformations with smallest weighted \(L^p\) average distortion and Nitsche type phenomena. J. Lond. Math. Soc. (2) 85(2), 282–300 (2012)Google Scholar
  31. 31.
    Jäger, W., Kaul, H.: Rotationally symmetric harmonic maps from a ball into a sphere and the regularity problem for weak solutions of elliptic systems. J. Reine Angew. Math. 343, 146–161 (1983)zbMATHGoogle Scholar
  32. 32.
    Meynard, F.: Existence and nonexistence results on the radially symmetric cavitation problem. Quart. Appl. Math. 50, 201–226 (1992)CrossRefzbMATHGoogle Scholar
  33. 33.
    Müller, S., Spector, S.J.: An existence theory for nonlinear elasticity that allows for cavitation. Arch. Ration. Mech. Anal. 131, 1–66 (1995)CrossRefzbMATHGoogle Scholar
  34. 34.
    Nitsche, J.C.C.: On the modulus of doubly connected regions under harmonic mappings. Am. Math. Monthly 69, 781–782 (1962)CrossRefzbMATHGoogle Scholar
  35. 35.
    Reshetnyak, YuG: Space Mappings with Bounded Distortion. American Mathematical Society, Providence (1989)CrossRefzbMATHGoogle Scholar
  36. 36.
    Sivaloganathan, J.: Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity. Arch. Ration. Mech. Anal. 96, 97–136 (1986)CrossRefzbMATHGoogle Scholar
  37. 37.
    Sivaloganathan, J., Spector, S.J.: Necessary conditions for a minimum at a radial cavitating singularity in nonlinear elasticity. Ann. Inst. Henri Poincaré Anal. Non Linéaire 25(1), 201–213 (2008)Google Scholar
  38. 38.
    Sivaloganathana, J., Spector, S.J.: On irregular weak solutions of the energy-momentum equations. Proc. R. Soc. Edinb. A 141, 193–204 (2011)CrossRefGoogle Scholar
  39. 39.
    Stuart, C.A.: Radially symmetric cavitation for hyperelastic materials. Anal. Non Liné aire 2, 33–66 (1985)CrossRefzbMATHGoogle Scholar
  40. 40.
    Šverák, V.: Regularity properties of deformations with finite energy. Arch. Ration. Mech. Anal. 100(2), 105–127 (1988)CrossRefzbMATHGoogle Scholar
  41. 41.
    Turowski, G.: Behaviour of doubly connected minimal surfaces at the edges of the support surface. Arch. Math. (Basel) 77(3), 278–288 (2001)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of JyväskyläJyväskyläFinland
  2. 2.Department of MathematicsSyracuse UniversitySyracuseUSA

Personalised recommendations