Deformations of finite conformal energy: Boundary behavior and limit theorems

Arch Rational Mech Anal

2015

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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.

Section: Applications To Thin Plates and Filmsmentioning

confidence: 98%

Monotone Sobolev Mappings of Planar Domains and Surfaces

Iwaniec

Arch Rational Mech Anal

2015

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“…It is known that a homeomorphism h : X onto − − → Y between Lipschitz domains in the Sobolev space W 1,2 (X, Y) extends continuously up to the boundaries [28]. And it is topologically clear that a continuous extension of a homeomorphism h : X onto − − → Y results in monotone mappings h : X onto − − → Y and h : ∂X onto − − → ∂Y .…”

Section: The Dirichlet Energy the Dirichlet Energy (Or Dirichlet Intmentioning

confidence: 99%

“…Recently there have been new challenges and substantial work done [3,4,5,20,28,29] on minimizing the Dirichlet energy integral…”

Section: Introductionmentioning

confidence: 99%

Mappings of Least Dirichlet Energy and their Hopf Differentials

Iwaniec

Arch Rational Mech Anal

2013

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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.

“…For example, no quasiconformal mapping of a ball can produce a spike which points into the ball [17]. It would be interesting to describe all pairs of the domains X and Y for which there exist deformations h : X [31]. Precisely, we require that ∂Y consists of at least two finitely many boundary components.…”

Section: N-harmonic Integralsmentioning

confidence: 99%

“…Ball's fundamental paper [5] accounts for the principles of hyperelasticity and sets up mathematical problems. In presenting the recent advances we have relied on a few new existing articles [2,3,28,29,30,31,32,33]. …”

mentioning

confidence: 99%

An Invitation to n-Harmonic Hyperelasticity

Iwaniec¹,

Pure and Applied Mathematics Quarterly

2011

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Abstract:We give an account of some recent developments in which quasiconformal theory and nonlinear elasticity share common problems of compelling mathematical interest. Geometric function theory is currently a field of enormous activity where the language and general framework of nonlinear elasticity is extremely fruitful and significant. As this interplay developed n-harmonic deformations became valid and well acknowledged as generalization of conformal mappings in R n . We have also found a place for n-harmonic deformations in the theory of hyperelasticity. J. Ball's fundamental paper [5] accounts for the principles of hyperelasticity and sets up mathematical problems. In presenting the recent advances we have relied on a few new existing articles [2,3,28,29,30,31,32,33].