Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T17:35:53.208Z Has data issue: false hasContentIssue false
  • English
  • Français

Radial symmetry of minimizers to the weighted Dirichlet energy

Part of: Elliptic equations and systems Geometric function theory

Published online by Cambridge University Press:  20 February 2020

Aleksis Koski  and
Jani Onninen
Aleksis Koski
Affiliation:
Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FI-40014, Finland (aleksis.t.koski@jyu.fi)
Jani Onninen
Affiliation:
Department of Mathematics, Syracuse University, Syracuse, NY13244, USA (jkonnine@syr.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the problem of minimizing the weighted Dirichlet energy between homeomorphisms of planar annuli. A known challenge lies in the case when the weight λ depends on the independent variable z. We prove that for an increasing radial weight λ(z) the infimal energy within the class of all Sobolev homeomorphisms is the same as in the class of radially symmetric maps. For a general radial weight λ(z) we establish the same result in the case when the target is conformally thin compared to the domain. Fixing the admissible homeomorphisms on the outer boundary we establish the radial symmetry for every such weight.

Keywords

Variational integralsHarmonic mappingsEnergy-minimal deformations

MSC classification

Primary: 35J60: Nonlinear elliptic equations
Secondary: 30C70: Extremal problems for conformal and quasiconformal mappings, variational methods
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 1 , February 2021 , pp. 169 - 186
Check for updates
Copyright
Copyright © The Author(s) 2020. Published by The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Astala, K., Iwaniec, T. and Martin, G.. Elliptic partial differential equations and quasiconformal mappings in the plane (Princeton, NJ: Princeton University Press, 2009).CrossRefGoogle Scholar
2Astala, K., Iwaniec, T. and Martin, G.. Deformations of annuli with smallest mean distortion. Arch. Ration. Mech. Anal. 195 (2010), 899–921.CrossRefGoogle Scholar
3Astala, K., Iwaniec, T., Martin, G. and Onninen, J.. Extremal mappings of finite distortion. Proc. London Math. Soc. 91 (2005), 655–702.CrossRefGoogle Scholar
4Ball, J. M.. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63 (1976), 337–403.CrossRefGoogle Scholar
5Ball, J. M.. Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos. Trans. R. Soc. London A 306 (1982), 557–611.Google Scholar
6Ball, J. M.. Singularities and computation of minimizers for variational problems. Foundations of Computational Mathematics (Oxford, 1999), London Math. Soc. Lecture Note Ser., 284, pp. 1–20 (Cambridge: Cambridge University Press, 2001).Google Scholar
7Ball, J. M.. Some open problems in elasticity. Geometry, mechanics, and dynamics, pp. 3–59 (New York: Springer, 2002).Google Scholar
8Ball, J. M.. Progress and puzzles in nonlinear elasticity. Proceedings of course on Poly-, Quasi- and Rank-One Convexity in Applied Mechanics (Udine: CISM, 2010).CrossRefGoogle Scholar
9Ball, J. M., Currie, J. C. and Olver, P. J.. Null Lagrangians, weak continuity, and variational problems of arbitrary order. J. Funct. Anal. 41 (1981), 135–174.CrossRefGoogle Scholar
10Chen, S. and Kalaj, D.. Total energy of radial mappings. Nonlinear Anal. 167 (2018), 21–28.CrossRefGoogle Scholar
11Coron, J.-M. and Gulliver, R. D.. Minimizing p-harmonic maps into spheres. J. Reine Angew. Math. 401 (1989), 82–100.Google Scholar
12Edelen, D. G. B.. The null set of the Euler-Lagrange operator. Arch. Ration. Mech. Anal. 11 (1962), 117–121.CrossRefGoogle Scholar
13Hardt, R., Lin, F. H. and Wang, C. Y.. The p-energy minimality of $x/|x|$. Comm. Anal. Geom. 6 (1998), 141–152.CrossRefGoogle ScholarPubMed
14Hencl, S. and Koskela, P.. Regularity of the inverse of a planar Sobolev homeomorphism. Arch. Ration. Mech. Anal. 180 (2006), 75–95.CrossRefGoogle Scholar
15Hencl, S. and Koskela, P.. Lectures on mappings of finite distortion. Lecture Notes in Mathematics, vol. 2096 (Cham: Springer, 2014).CrossRefGoogle Scholar
16Hencl, S., Koskela, P. and Onninen, J.. A note on extremal mappings of finite distortion. Math. Res. Lett. 12 (2005), 231–237.CrossRefGoogle Scholar
17Hong, M.-C.. On the minimality of the p-harmonic map $\frac {x}{|x|} \colon B^n \to S^{n-1}$. Calc. Var. Partial Differ. Equ. 13 (2001), 459–468.Google Scholar
18Iwaniec, T.. On the concept of the weak Jacobian and Hessian. Rep. Univ. Jyväskylä Dept. Math. Stat. 83 (2001), 181–205.Google Scholar
19Iwaniec, T., Kovalev, L. V. and Onninen, J.. The Nitsche conjecture. J. Amer. Math. Soc. 24 (2011), 345–373.CrossRefGoogle Scholar
20Iwaniec, T., Kovalev, L. V. and Onninen, J.. Doubly connected minimal surfaces and extremal harmonic mappings. J. Geom. Anal. 22 (2012), 726–762.CrossRefGoogle Scholar
21Iwaniec, T., Kovalev, L. V. and Onninen, J.. Hopf differentials and smoothing Sobolev homeomorphisms. Int. Math. Res. Not. (IMRN) 2012 (2012), 3256–3277.Google Scholar
22Iwaniec, T. and Martin, G.. Geometric function theory and non-linear analysis. Oxford Mathematical Monographs (New York: Oxford University Press, 2001).Google Scholar
23Iwaniec, T. and Onninen, J.. Hyperelastic deformations of smallest total energy. Arch. Ration. Mech. Anal. 194 (2009), 927–986.CrossRefGoogle Scholar
24Iwaniec, T. and Onninen, J.. Neohookean deformations of annuli, existence, uniqueness and radial symmetry. Math. Ann. 348 (2010), 35–55.CrossRefGoogle Scholar
25Iwaniec, T. and Onninen, J.. n-Harmonic mappings between annuli. Mem. Amer. Math. Soc. 218 (2012), 1023.Google Scholar
26Iwaniec, T. and Onninen, J.. Sobolev mappings and variational integrals in geometric function theory, book in progress).Google Scholar
27Iwaniec, T. and Onninen, J.. Monotone Sobolev mappings of planar domains and surfaces. Arch. Ration. Mech. Anal. 219 (2016), 159–181.CrossRefGoogle Scholar
28Iwaniec, T. and Onninen, J.. Limits of Sobolev homeomorphisms. J. Eur. Math. Soc. (JEMS) 19 (2017), 473–505.CrossRefGoogle Scholar
29Iwaniec, T., Onninen, J. and Radice, T.. The Nitsche phenomenon for weighted Dirichlet energy. Adv. Calc. Var., to appear.Google Scholar
30Jordens, M. and Martin, G. J.. Deformations with smallest weighted L p average distortion and Nitsche type phenomena. J. Lond. Math. Soc. 85 (2012), 282–300.Google Scholar
31Jäger, W. and 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 (1983), 146–161.Google Scholar
32Kalaj, D.. Deformations of annuli on Riemann surfaces and the generalization of Nitsche conjecture. J. London Math. Soc. 93 (2016), 683–702.CrossRefGoogle Scholar
33Kalaj, D.. On J. C. C. Nitsche type inequality for annuli on Riemann surfaces. Israel J. Math. 218 (2017), 67–81.CrossRefGoogle Scholar
34Koski, A. and Onninen, J.. Radial symmetry of p-harmonic minimizers. Arch. Rational Mech. Anal., to appear.Google Scholar
35Meynard, F.. Existence and nonexistence results on the radially symmetric cavitation problem. Quart. Appl. Math. 50 (1992), 201–226.CrossRefGoogle Scholar
36Morrey, C. B.. The topology of (path) surfaces. Amer. J. Math. 57 (1935), 17–50.CrossRefGoogle Scholar
37Müller, S.. Higher integrability of determinants and weak convergence in L1. J. Reine Angew. Math. 412 (1990), 20–34.Google Scholar
38Müller, S. and Spector, S. J.. An existence theory for nonlinear elasticity that allows for cavitation. Arch. Ration. Mech. Anal. 131 (1995), 1–66.CrossRefGoogle Scholar
39Reshetnyak, Yu. G.. Certain geometric properties of functions and mappings with generalized derivatives. (Russian) Sibirsk. Mat. Ž. 7 (1966), 886–919.Google Scholar
40Reshetnyak, Yu. G.. Space mappings with bounded distortion (Providence, RI: American Mathematical Society, 1989).CrossRefGoogle Scholar
41Rickman, S.. Quasiregular mappings (Berlin: Springer-Verlag, 1993).CrossRefGoogle Scholar
42Sivaloganathan, J.. Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity. Arch. Ration. Mech. Anal. 96 (1986), 97–136.CrossRefGoogle Scholar
43Sivaloganathan, J. and Spector, S. J.. Necessary conditions for a minimum at a radial cavitating singularity in nonlinear elasticity. Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), 201–213.CrossRefGoogle Scholar
44Stuart, C. A.. Radially symmetric cavitation for hyperelastic materials. Anal. Non Liné aire 2 (1985), 33–66.CrossRefGoogle Scholar