Mappings of finite distortion: Monotonicity and continuity

Iwaniec, Tadeusz; Koskela, Pekka; Onninen, Jani

doi:10.1007/s002220100130

Cited by 178 publications

(170 citation statements)

References 23 publications

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“…Thus it only remains to justify the uniform estimates (2.7). Local estimates like this are well known for monotone mappings of Sobolev class W 1,2 (X , Y) [21,26]. The global inequality follows by the standard method of extending h beyond the boundaries.…”

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

confidence: 86%

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: The Dirichlet Energy the Dirichlet Energy (Or Dirichlet Intmentioning

confidence: 86%

Mappings of Least Dirichlet Energy and their Hopf Differentials

Iwaniec

Onninen

2013

Arch Rational Mech Anal

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“…Under the assumptions f ∈ W 1,n loc and K ∈ L n−1+ε loc it is known that the mapping f of finite distortion is continuous, discrete and open and satisfies the condition N and that distributional Jacobian of f is equal to the pointwise Jacobian of f (see [10], [18], [28], [24], [25] and [20]). …”

Section: Applications To Mappings Of Finite Distortionmentioning

confidence: 99%

Mappings of finite distortion: Reverse inequalities for the Jacobian

2007

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“…The grand Lebesgue spaces were introduced by Iwaniec and Sbordone [12] and they found many applications in Analysis, see [12], [10]. The small Lebesgue spaces were introduced by A. Fiorenza [5] as associate to grand spaces.…”

Section: Introductionmentioning

confidence: 99%