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Radial Symmetry of p-Harmonic Minimizers

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Abstract

“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.

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References

  1. Adamowicz, T.: On the Geometry of p-Harmonic Mappings, Ph.D. thesis, Syracuse University, 2008

  2. Antman, S.S.: Nonlinear Problems of Elasticity. Applied Mathematical Sciences, Vol. 107. Springer, New York, 1995

  3. Astala, K., Iwaniec, T., Martin, G.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton University Press, Princeton (2009)

    MATH  Google Scholar 

  4. Astala, K., Iwaniec, T., Martin, G.: Deformations of annuli with smallest mean distortion. Arch. Ration. Mech. Anal. 195(3), 899–921 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63(4), 337–403 (1976/1977)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ball, J.M.: Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos. Trans. R. Soc. Lond. A 306, 557–611 (1982)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  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, 1977

  9. Ball, J.M.: Existence of solutions in finite elasticity. Proceedings of the IUTAM Symposium on Finite Elasticity. Martinus Nijhoff, 1981

  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, 1984

  11. Ball, J.M.: Some open problems in elasticity. Geometry, mechanics, and dynamics. Springer, New York, 3–59, 2002

  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, 1956

  13. Ciarlet, P.G.: Mathematical elasticity Vol. I. Three-dimensional elasticity, Studies in Mathematics and its Applications, Vol. 20. North-Holland Publishing Co., Amsterdam, 1988

  14. Coron, J.-M., Gulliver, R.D.: Minimizing p-harmonic maps into spheres. J. Reine Angew. Math. 401, 82–100 (1989)

    MATH  MathSciNet  Google Scholar 

  15. Cuneo, D.: Mappings Between Annuli of Mimimal p-Harmonic Energy. Ph.D. thesis, Syracuse University, 2017

  16. Hardt, R., Lin, F.H., Wang, C.Y.: The p-energy minimality of \(x/|x|\). Commun. Anal. Geom. 6, 141–152 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hencl, S., Koskela, P.: Regularity of the inverse of a planar Sobolev homeomorphism. Arch. Ration. Mech. Anal. 1 80(1), 75–95 (2006)

  18. Hencl, S., Pratelli, A.: Diffeomorphic Approximation of \(W^{1,1}\) Planar Sobolev Homeomorphisms. J. Eur. Math. Soc. (to appear)

  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)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  21. Iwaniec, T., Martin, G.: Geometric Function Theory and Non-linear Analysis. Oxford University Press, Oxford Mathematical Monographs (2001)

    MATH  Google Scholar 

  22. Iwaniec, T., Kovalev, L.V., Onninen, J.: The Nitsche conjecture. J. Am. Math. Soc. 24(2), 345–373 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  23. Iwaniec, T., Kovalev, L.V., Onninen, J.: Diffeomorphic approximation of Sobolev homeomorphisms. Arch. Ration. Mech. Anal. 201(3), 1047–1067 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  24. Iwaniec, T., Kovalev, L.V., Onninen, J.: Doubly connected minimal surfaces and extremal harmonic mappings. J. Geom. Anal. 22(3), 726–762 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  25. Iwaniec, T., Onninen, J.: Hyperelastic deformations of smallest total energy. Arch. Ration. Mech. Anal. 194(3), 927–986 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Iwaniec, T., Onninen, J.: Neohookean deformations of annuli, existence, uniqueness and radial symmetry. Math. Ann. 348(1), 35–55 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  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, 319–343

  28. Iwaniec, T., Onninen, J.: \(n\)-Harmonic mappings between annuli Mem. Am. Math. Soc. 218 (2012)

  29. Iwaniec, T., Onninen, J.: Limits of Sobolev homeomorphisms. J. Eur. Math. Soc. (JEMS) 19(2), 473–505 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  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)

    MATH  MathSciNet  Google Scholar 

  32. Meynard, F.: Existence and nonexistence results on the radially symmetric cavitation problem. Quart. Appl. Math. 50, 201–226 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  33. Müller, S., Spector, S.J.: An existence theory for nonlinear elasticity that allows for cavitation. Arch. Ration. Mech. Anal. 131, 1–66 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  34. Nitsche, J.C.C.: On the modulus of doubly connected regions under harmonic mappings. Am. Math. Monthly 69, 781–782 (1962)

    Article  MATH  Google Scholar 

  35. Reshetnyak, YuG: Space Mappings with Bounded Distortion. American Mathematical Society, Providence (1989)

    Book  MATH  Google Scholar 

  36. Sivaloganathan, J.: Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity. Arch. Ration. Mech. Anal. 96, 97–136 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  38. Sivaloganathana, J., Spector, S.J.: On irregular weak solutions of the energy-momentum equations. Proc. R. Soc. Edinb. A 141, 193–204 (2011)

    Article  MathSciNet  Google Scholar 

  39. Stuart, C.A.: Radially symmetric cavitation for hyperelastic materials. Anal. Non Liné aire 2, 33–66 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  40. Šverák, V.: Regularity properties of deformations with finite energy. Arch. Ration. Mech. Anal. 100(2), 105–127 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  41. Turowski, G.: Behaviour of doubly connected minimal surfaces at the edges of the support surface. Arch. Math. (Basel) 77(3), 278–288 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), 40014, Jyväskylä, Finland

    Aleksis Koski & Jani Onninen

  2. Department of Mathematics, Syracuse University, Syracuse, NY, 13244, USA

    Jani Onninen

Authors
  1. Aleksis Koski
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  2. Jani Onninen
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Correspondence to Jani Onninen.

Additional information

Communicated by L. Székelyhidi

A. Koski was supported by the ERC Starting Grant No. 307023. J. Onninen was supported by the NSF Grant DMS-1700274.

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Koski, A., Onninen, J. Radial Symmetry of p-Harmonic Minimizers. Arch Rational Mech Anal 230, 321–342 (2018). https://doi.org/10.1007/s00205-018-1246-0

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