TY - JOUR

T1 - Deformations of finite conformal energy

T2 - Existence and removability of singularities

AU - Iwaniec, Tadeusz

AU - Onninen, Jani

N1 - Funding Information:
Received 27 July 2008; revised 10 February 2009; published online 14 May 2009. 2000 Mathematics Subject Classification 30C60, 35J15, 35J70 (Primary). Iwaniec was supported by the NSF grant DMS-0800416, and Onninen by the NSF grant DMS-0701059. †When p < n, the topology can inhibit the space C∞(X, Y) from being dense in W 1,p(X, Y); see [6, 21, 22].

PY - 2010/1

Y1 - 2010/1

N2 - This paper features a class of mappings h = (h1,...,h n) :X→ontoY between bounded domains X,Y ⊂ ℝn, having finite n-harmonic energy, such that we have ℰ[h]=∫x ∥Dh(x)∥ndx, ∥Dh∥ 2=Tr(D*h Dh). The fundamental question is whether or not the domains X, Y ⊂ ℝn of the same topological type admit a homeomorphism h: X → onto Y in a given homotopy class having finite energy. The examples of non-existence, somewhat testing our theory, arise when we remove from bounded smooth domains X and Y thin subsets x ⊂ X and γ ⊂ Y, referred to as cracks or fractures. We are looking for homeomorphisms h: X\x → ontoY\γ of finite energy for which γis the cluster set of h over x. In general, infinite energy is required in order to increase the dimension of a crack x ⊂ X;that is, when dim x < dim γ ≤ n-1. Suppose now that a bounded deformation h: X\x → ℝn of finite energy is given. Does h extend continuously to X and, if so, is the extension injective on X. We give affirmative answers to these questions.

AB - This paper features a class of mappings h = (h1,...,h n) :X→ontoY between bounded domains X,Y ⊂ ℝn, having finite n-harmonic energy, such that we have ℰ[h]=∫x ∥Dh(x)∥ndx, ∥Dh∥ 2=Tr(D*h Dh). The fundamental question is whether or not the domains X, Y ⊂ ℝn of the same topological type admit a homeomorphism h: X → onto Y in a given homotopy class having finite energy. The examples of non-existence, somewhat testing our theory, arise when we remove from bounded smooth domains X and Y thin subsets x ⊂ X and γ ⊂ Y, referred to as cracks or fractures. We are looking for homeomorphisms h: X\x → ontoY\γ of finite energy for which γis the cluster set of h over x. In general, infinite energy is required in order to increase the dimension of a crack x ⊂ X;that is, when dim x < dim γ ≤ n-1. Suppose now that a bounded deformation h: X\x → ℝn of finite energy is given. Does h extend continuously to X and, if so, is the extension injective on X. We give affirmative answers to these questions.

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U2 - 10.1112/plms/pdp016

DO - 10.1112/plms/pdp016

M3 - Article

AN - SCOPUS:73649114925

VL - 100

SP - 1

EP - 23

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

SN - 0024-6115

IS - 1

ER -