Limits of Sobolev homeomorphisms

Iwaniec, Tadeusz; Onninen, Jani

doi:10.4171/jems/671

Cited by 20 publications

(16 citation statements)

References 43 publications

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“…Thus, in particular, Theorem 1.3 implies that h can be realized as W 1,p -strong limit of homeomorphisms. We therefore recover a result in [31]; accordingly, weak sequential closure and strong closure of H p (X, Y) , p 2 , are the same.…”

Section: Introductionsupporting

confidence: 79%

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Monotone Sobolev Mappings of Planar Domains and Surfaces

Iwaniec

Onninen

2015

Arch Rational Mech Anal

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Abstract. An approximation theorem of Youngs (1948) asserts that a continuous map between compact oriented topological 2-manifolds (surfaces) is monotone if and only if it is a uniform limit of homeomorphisms. Analogous approximation of Sobolev mappings is at the very heart of Geometric Function Theory (GFT) and Nonlinear Elasticity (NE). In both theories the mappings in question arise naturally as weak limits of energy-minimizing sequences of homeomorphisms. As a result of this, the energy-minimal mappings turn out to be monotone. In the present paper we show that, conversely, monotone mappings in the Sobolev space W 1,p , 1 < p < ∞ , are none other than W 1,p -weak (also strong) limits of homeomorphisms. In fact, these are limits of diffeomorphisms. By way of illustration, we establish the existence of energy-minimal deformations within the class of Sobolev monotone mappings for p -harmonic type energy integrals.

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Section: Introductionsupporting

confidence: 79%

“…Concerning the positive sign of the Jacobian determinant of F , we shall appeal to a p-harmonic variant of Hurwitz Theorem [31,Theorem 4.9]; that is, Theorem 3.6. If a sequence F n : Ω → R 2 of p-harmonic (coordinate-wise) orientation preserving diffeomorphisms converges uniformly to…”

Section: Lemma 34mentioning

confidence: 99%

Monotone Sobolev Mappings of Planar Domains and Surfaces

Iwaniec

Onninen

2015

Arch Rational Mech Anal

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“…In the next lemma, \BbbX and \BbbY are \ell -connected Lipschitz domains in \BbbR 2 ; see [26,Lemma 4.3].…”

Section: Modulus Of Continuity and Conformal Energymentioning

confidence: 99%

A Neohookean Model of Plates

Iwaniec¹,

Onninen²,

Pankka³

et al. 2021

SIAM J. Math. Anal.

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This article is about hyperelastic deformations of plates (planar domains) which minimize a neohookean-type energy. Particularly, we investigate a stored energy functional introduced by J. M. Ball [Proc. Roy. Soc. Edinb. Sect. A, 88 (1981), pp. 315--328]. The mappings under consideration are Sobolev homeomorphisms and their weak limits. They are monotone in the sense of C. B. Morrey. One major advantage of adopting monotone Sobolev mappings lies in the existence of the energy-minimal deformations. However, injectivity is inevitably lost, so an obvious question to ask is, what are the largest subsets of the reference configuration on which minimal deformations remain injective? The fact that such subsets have full measure should be compared with the notion of global invertibility, which deals with subsets of the deformed configuration instead. In this connection we present a Cantor-type construction to show that both the branch set and its image may have positive area. Another novelty of our approach lies in allowing the elastic deformations to be free along the boundary, known as frictionless problems.

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“…Specifically, the function m p : (1, ∞] → (1, ∞] is given by the formula (2.10). To formulate Theorem 1.3 in its full generality, we need to adopt the limits of Sobolev homeomorphisms as legitimate deformations [29]. We denote the class of strong W 1,p -limits of sequences in H 1,p Id (A, A * ) by H…”

Section: Theorem 13 [Fixed Boundary Values] Letmentioning

confidence: 99%

Radial Symmetry of p-Harmonic Minimizers

Koski

Onninen

2018

Arch Rational Mech Anal

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It is still not known if the radial cavitating minimizers obtained by Ball [J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Phil. Trans. R. Soc. Lond. A 306 (1982) 557-611] (and subsequently by many others) are global minimizers of any physically reasonable nonlinearly elastic energy". The quotation is from [37] 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.− − → A * in W 1,p (A, R n ) which are furthermore assumed to preserve the order of the 2010 Mathematics Subject Classification. Primary 35J60; Secondary 30C70.

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Limits of Sobolev homeomorphisms

Cited by 20 publications

References 43 publications

Monotone Sobolev Mappings of Planar Domains and Surfaces

Monotone Sobolev Mappings of Planar Domains and Surfaces

A Neohookean Model of Plates

Radial Symmetry of p-Harmonic Minimizers

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