Skip to main content Skip to main navigation menu Skip to site footer
  • Login
  • Language
    • English
    • Język Polski
  • Menu
  • Home
  • Current
  • Online First
  • Archives
  • About
    • About the Journal
    • Submissions
    • Editorial Team
    • Privacy Statement
    • Contact
  • Login
  • Language:
  • English
  • Język Polski

Topological Methods in Nonlinear Analysis

Topology of twists, extremising twist paths and multiple solutions to the nonlinear system in variation $\mathscr{L}[u] = \nabla \mathscr{P}$
  • Home
  • /
  • Topology of twists, extremising twist paths and multiple solutions to the nonlinear system in variation $\mathscr{L}[u] = \nabla \mathscr{P}$
  1. Home /
  2. Archives /
  3. Vol 54, No 2 (December 2019) /
  4. Articles

Topology of twists, extremising twist paths and multiple solutions to the nonlinear system in variation $\mathscr{L}[u] = \nabla \mathscr{P}$

Authors

  • George Morrison
  • Ali Taheri

Keywords

Nonlinear elliptic systems, incompressible twists, multiple solutions, restricted energies, closed geodesics on SO(n), self-maps of spheres, Hopf degree

Abstract

In this paper we address questions on the existence and multiplicity of a class of geometrically motivated mappings with certain symmetries that serve as solutions to the nonlinear system in variation: $$ \rom{ELS} [(u, \mathscr{P}), \Omega] = \begin{cases} [\nabla u]^t \rom{div} [F_\xi\nabla u ] -F_s [\nabla u]^t u = \nabla \mathscr{P} &\text{in } \Omega, \\ \det \nabla u = 1 &\text{in } \Omega, \\ u \equiv x &\text{on } \partial \Omega. \end{cases} $$% Here $\Omega \subset \R^n$ is a bounded domain, $F=F(r, s, \xi)$ is a sufficiently smooth Lagrangian, $F_s=F_s(|x|, |u|^2, |\nabla u|^2)$ and $F_\xi=F_\xi(|x|, |u|^2, |\nabla u|^2)$ with $F_s$ and $F_\xi$ denoting the derivatives of $F$ with respect to the second and third variables respectively while $\mathscr{P}$ is an {\it a priori} unknown hydrostatic pressure resulting from the incompressibility constraint $\det \nabla u =1$. Among other things, by considering twist mappings $u$ with an SO$(n)$-valued twist path, we prove the existence of multiple and topologically distinct solutions to ELS for $n \ge 2$ even versus the only ({\it non}) twisting solution $u \equiv x$ for $n \ge 3$ odd. An extremality analysis for twist paths and those of Lie exponential types and a suitable formulation of a differential operator action on twists relating to ELS are the key ingredients in the proof.

References

S.S. Antman, Nonlinear Problems of Elasticity, Applied Mathematical Sciences, vol. 107, 2nd edition, Springer, 2005.

K. Astala, T. Iwaniec, G. Martin and J. Onninen, Extremal mappings of finite distortion, Proc. Lond. Math. Soc. 91 (2005), 655–702.

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

J.M. Ball, Some open problems in elasticity, Geometry, Mechanics and Dynamics, Springer, New York, 2002, pp. 3–59.

J.M. Ball and D.G. Schaeffer, Bifurcation and stability of homogeneous equilibrium configurations of an elastic body under dead-load traction, Math. Proc. Camb. Phil. Soc. 94 (1983], 315–339.

J.S. Birman, Braids, links and mapping class groups, Ann. of Math. Stud. 82, Princeton University Press, 1975.

L. Cesari, Optimization Theory and Applications: Problems with Ordinary Differential Equations, Applications of Mathematics, vol. 17, Springer, 1983.

P. Ciarlet, Mathematical Elasticity: Three Dimensional Elasticity, vol. 1, Elsevier, 1988.

B. Dacorogna, Direct Methods in the Calculus of Variations, 2nd edition, Applied Mathematical Sciences, vol. 78, Springer, 2007.

S. Day and A. Taheri, A class of extremising sphere-valued maps with inherent maximal tori symmetries in SO(n), Bound. Value Prob. 187 (2017).

S. Day and A. Taheri, Geodesics on SO(n) and a class of spherically symmetric maps as solutions to a nonlinear generalised harmonic map problem, Topol. Methods Nonlinear Anal. 51 (2018), 637–662.

L.C. Evans and R.F. Gariepy, On the partial regularity of energy-minimizing, area preserving maps, Calc. Var. Partial Differential Equations 63 (1999), 357–372.

V.L. Hansen, The homotopy problem for the components in the space of maps on the n-sphere, Quart. J. Math. 25 (1974), 313–321.

V.L. Hansen, The homotopy groups of a space of maps between oriented closed surfaces, Bull. Lond. Math. Soc. 15 (1983), 360–364.

S. Hencl and P. Koskela, Lectures on Mappings of Finite Distortion, Lecture Notes in Mathematics, vol. 2096, Springer, 2014.

T. Iwaniec and J. Onninen, n-harmonic mappings between annuli: the art of integrating free Lagrangians, Mem. Amer. Math. Soc. 218 (2012).

F. John, Remarks on the non-linear theory of elasticity, Semin. Ist. Naz. Alta Mat., Ediz. Cremonese, Rome, 1962/1963, pp. 474–482.

F. John, Uniqueness of non-linear elastic equilibrium for prescribed boundary displacements and sufficiently small strains, Comm. Pure Appl. Math. 25 (1972), 617–634.

A.W. Knapp, Lie Groups Beyond an Introduction, Progress in Mathematics, 2nd edition, Birkhäuser, 2002.

R.J. Knops and C.A. Stuart, Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity, Arch. Rational Mech. Anal. 86 (1984), 233–249.

S.S. Koh, Note on the properties of the components of the mapping spaces X S , Proc. Amer. Math. Soc. 11 (1960), 896–904.

C.B. Morrey, Multiple Integrals in the Calculus of Variations, Classics in Mathematics, vol. 130, Springer, 1966.

C. Morris and A. Taheri, Twist mappings as energy minimisers in homotopy classes: symmetrisation and the coarea formula, Nonlinear Anal. 152 (2017), 250–275.

C. Morris and A. Taheri, Annular rearrangements, incompressible axi-symmetric whirls and L1 -local minimisers of the distortion energy, Nonlin. Diff. Equ. Appl. 25 (2018).

K. Post and J. Sivaloganathan, On homotopy conditions and the existence of multiple equilibria in finite elasticity, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), 595–614.

Y.G. Reshetnyak, Space Mappings with Bounded Distortion, Transl. Math. Monographs, Vol. 73, Amer. Math. Soc., 1989.

M.S. Shahrokhi-Dehkordi and A. Taheri, Generalised twists, stationary loops, and the Dirichlet energy over a space of measure preserving maps, Cal. Var. Partial Differential Equations 35 (2009), 191–213.

M.S. Shahrokhi-Dehkordi and A. Taheri, Generalised twists, SO(n), and the p-energy over a space of measure-preserving maps, Ann. Inst. H. Poincaré 26 (2009), 1897–1924.

M.S. Shahrokhi-Dehkordi and A. Taheri, Quasiconvexity and uniqueness of stationary points on a space of measure preserving maps, J. Conv. Anal. 17 (2010), 69–79.

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

A. Taheri, Local minimizers and quasiconvexity – the impact of topology, Arch. Rational Mech. Anal. 176 (2005), 363-414.

A. Taheri, Minimizing the Dirichlet energy over a space of measure preserving maps, Topol. Methods Nonlinear Anal. 33 (2009), 179–204.

A. Taheri, Homotopy classes of self-maps of annuli generalised twists and spin degree, Arch. Rational Mech. Anal. 197 (2010), 239–270.

A. Taheri, Spherical twists, stationary paths and harmonic maps from generalised annuli into spheres, Nonlin. Diff. Equ. Appl. 19 (2012), 79–95.

A. Taheri, Function Spaces and Partial Differential Equations, Vols. I and II, Oxford Lecture Series in Mathematics and its Applications, OUP, 2015.

S.K. Vodopyanov and V.M. Gol’dshtein, Quasiconformal mappings and spaces of functions with generalized first derivatives, Siberian Math. J. 17 (1977), 515–531.

G.W. Whitehead, On homotopy groups of spheres and rotation groups, Ann. of Math. 43 (1942), 634–640.

G.W. Whitehead, On products in homotopy groups, Ann. of Math. 47 (1946), 460–475.

G.W. Whitehead, Elements of Homotopy Theory, Springer–Verlag, 1978.

J.H.C. Whitehead, On certain theorems of G.W. Whitehead, Ann. of Math. 58 (1953), 418–428.

Downloads

  • PREVIEW
  • FULL TEXT

Published

2019-11-09

How to Cite

1.
MORRISON, George and TAHERI, Ali. Topology of twists, extremising twist paths and multiple solutions to the nonlinear system in variation $\mathscr{L}[u] = \nabla \mathscr{P}$. Topological Methods in Nonlinear Analysis. Online. 9 November 2019. Vol. 54, no. 2, pp. 833 - 862. [Accessed 2 July 2025].
  • ISO 690
  • ACM
  • ACS
  • APA
  • ABNT
  • Chicago
  • Harvard
  • IEEE
  • MLA
  • Turabian
  • Vancouver
Download Citation
  • Endnote/Zotero/Mendeley (RIS)
  • BibTeX

Issue

Vol 54, No 2 (December 2019)

Section

Articles

Stats

Number of views and downloads: 0
Number of citations: 0

Search

Search

Browse

  • Browse Author Index
  • Issue archive

User

User

Current Issue

  • Atom logo
  • RSS2 logo
  • RSS1 logo

Newsletter

Subscribe Unsubscribe
Up

Akademicka Platforma Czasopism

Najlepsze czasopisma naukowe i akademickie w jednym miejscu

apcz.umk.pl

Partners

  • Akademia Ignatianum w Krakowie
  • Akademickie Towarzystwo Andragogiczne
  • Fundacja Copernicus na rzecz Rozwoju Badań Naukowych
  • Instytut Historii im. Tadeusza Manteuffla Polskiej Akademii Nauk
  • Instytut Kultur Śródziemnomorskich i Orientalnych PAN
  • Instytut Tomistyczny
  • Karmelitański Instytut Duchowości w Krakowie
  • Ministerstwo Kultury i Dziedzictwa Narodowego
  • Państwowa Akademia Nauk Stosowanych w Krośnie
  • Państwowa Akademia Nauk Stosowanych we Włocławku
  • Państwowa Wyższa Szkoła Zawodowa im. Stanisława Pigonia w Krośnie
  • Polska Fundacja Przemysłu Kosmicznego
  • Polskie Towarzystwo Ekonomiczne
  • Polskie Towarzystwo Ludoznawcze
  • Towarzystwo Miłośników Torunia
  • Towarzystwo Naukowe w Toruniu
  • Uniwersytet im. Adama Mickiewicza w Poznaniu
  • Uniwersytet Komisji Edukacji Narodowej w Krakowie
  • Uniwersytet Mikołaja Kopernika
  • Uniwersytet w Białymstoku
  • Uniwersytet Warszawski
  • Wojewódzka Biblioteka Publiczna - Książnica Kopernikańska
  • Wyższe Seminarium Duchowne w Pelplinie / Wydawnictwo Diecezjalne „Bernardinum" w Pelplinie

© 2021- Nicolaus Copernicus University Accessibility statement Shop