The Nitsche conjecture

Iwaniec, Tadeusz; Kovalev, Leonid V.; Onninen, Jani

doi:10.1090/s0894-0347-2010-00685-6

Cited by 56 publications

(50 citation statements)

References 33 publications

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“…Nevertheless, nonexistence of harmonic homeomorphisms between annuli X and Y in Case 2 was conjectured by J. C. C. Nitsche [45], [46] , prominent conjecture indeed. After several efforts in [17,33,39,38,40,51], the conjecture was finally confirmed in [19]. Similar cracking phenomena are observed in higher dimensions [27], see [31] for the L p -setting.…”

Section: Hopf-laplace Equationmentioning

confidence: 53%

Mappings of Least Dirichlet Energy and their Hopf Differentials

Iwaniec

Onninen

2013

Arch Rational Mech Anal

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Abstract. The paper is concerned with mappings h : X onto − − → Y between planar domains having least Dirichlet energy. The existence and uniqueness (up to a conformal change of variables in X ) of the energyminimal mappings is established within the class H 2(X, Y) of strong limits of homeomorphisms in the Sobolev space W 1,2 (X, Y) , a result of considerable interest in the mathematical models of Nonlinear Elasticity. The inner variation (of the independent variable in X) leads to the Hopf differential hzhz dz ⊗ dz and its trajectories. For a pair of doubly connected domains, in which X has finite conformal modulus, we establish the following principle:A mapping h ∈ H 2(X, Y) is energy-minimal if and only if its Hopf-differential is analytic in X and real along ∂X. In general, the energy-minimal mappings may not be injective, in which case one observes the occurrence of cracks in X . Nevertheless, cracks are triggered only by the points in ∂Y where Y fails to be convex. The general law of formation of cracks reads as follows:Cracks propagate along vertical trajectories of the Hopf differential from ∂X toward the interior of X where they eventually terminate before making a crosscut.

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Section: Hopf-laplace Equationmentioning

confidence: 53%

Mappings of Least Dirichlet Energy and their Hopf Differentials

Iwaniec

Onninen

2013

Arch Rational Mech Anal

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“…This conjecture is solved recently in positive by Iwaniec, Kovalev and Onninen in [4]. Some partial results have been obtained previously by Lyzzaik [18], Weitsman [24] and the author [15].…”

mentioning

confidence: 74%

“…The Nitsche phenomena is rooted in the theory of doubly connected minimal surfaces (cf. [4]) and in the existence problem of maps between annuli of the complex plane, Riemann surface or Euclidean space with the least Dirichlet energy (cf. [8], [16], [21]).…”

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confidence: 99%

On J. C. C. Nitsche type inequality for annuli on Riemann surfaces

Kalaj

2017

Isr. J. Math.

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ABSTRACT. Assume that (N , ) and (M, ℘) are two Riemann surfaces with conformal metrics and ℘. We prove that if there is a harmonic homeomorphism between an annulus A ⊂ N with a conformal modulus Mod(A) and a geodesic annulus A℘(p, ρ1, ρ2) ⊂ M, then we have ρ2/ρ1 ≥ Ψ℘Mod(A) 2 + 1, where Ψ℘ is a certain positive constant depending on the upper bound of Gaussian curvature of the metric ℘. An application for the minimal surfaces is given.

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“…Some other generalizations have been done in [7]. The Nitsche conjecture for Euclidean harmonic mappings is settled recently in [3] by Iwaniec, Kovalev and Onninen, showing that, only radial harmonic mappings g(z) = e iα f (z), where f is defined in (1.2), which inspired the Nitsche conjecture, making the extremal distortion of rounded annuli. For some partial results toward the Nitsche conjecture and some other generalizations we refer to the papers [4], [8] and [11].…”

Section: This Means That This Function Makes the Maximal Distortion Omentioning

confidence: 99%

Polyharmonic mappings and J. C. C. Nitsche type inequalities

Kalaj¹,

Ponnusamy²

2014

Glas. Mat. Ser. III

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Abstract. In this paper a J. C. C. Nitsche type inequality for polyharmonic mappings between rounded annuli on the Euclidean space R d is considered. The case of radial biharmonic mappings between annuli on the complex plane and the corresponding inequality is studied in detail. IntroductionBy d we denote a positive integer and by R d we denote the Euclidean space. The unit sphere isBy A(r, R) = {x ∈ R d : r < |x| < R} we denote an annulus with inner and outer radii r and R, respectively. A mapping u is called radial if u(x) = |x|u(x/|x|). The bi-harmonic equation is

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The Nitsche conjecture

Cited by 56 publications

References 33 publications

Mappings of Least Dirichlet Energy and their Hopf Differentials

Mappings of Least Dirichlet Energy and their Hopf Differentials

On J. C. C. Nitsche type inequality for annuli on Riemann surfaces

Polyharmonic mappings and J. C. C. Nitsche type inequalities

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