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Deformations of finite conformal energy: existence and removability of singularities
Author(s) -
Iwaniec Tadeusz,
Onninen Jani
Publication year - 2010
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdp016
Subject(s) - mathematics , bounded function , injective function , cantor set , homeomorphism (graph theory) , conformal map , energy (signal processing) , gravitational singularity , homotopy , finite set , combinatorics , class (philosophy) , dimension (graph theory) , pure mathematics , mathematical physics , geometry , mathematical analysis , statistics , artificial intelligence , computer science
This paper features a class of mappings h = ( h 1 , … , h m ) : → onto between bounded domains , ⊂ ℝ n , having finite n ‐harmonic energy, such that we have E [ h ] =∫ X‖ D h ( x ) ‖ n d x , ‖ D h ‖ 2 = Tr ( D * h D h ) . The fundamental question is whether or not the domains , ⊂ ℝ n 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 and thin subsets X ⊂ Xand Υ ⊂ Y , referred to as cracks or fractures. We are looking for homeomorphisms h : X \ X → onto Y \ ϒ of finite energy for which ϒ is 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 X < dim Υ ⩽ n − 1 . Suppose now that a bounded deformation h : X ∖ X → R nof finite energy is given. Does h extend continuously to and, if so, is the extension injective on ? We give affirmative answers to these questions.