### Abstract

This paper features a class of mappings h = (h^{1},...,h ^{n}) :X→^{onto}Y between bounded domains X,Y ⊂ ℝ^{n}, having finite n-harmonic energy, such that we have ℰ[h]=∫_{x} ∥Dh(x)∥^{n}dx, ∥Dh∥ ^{2}=Tr(D*h Dh). The fundamental question is whether or not the domains X, Y ⊂ 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 X and Y thin subsets x ⊂ X and γ ⊂ Y, referred to as cracks or fractures. We are looking for homeomorphisms h: X\x → ^{onto}Y\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 → ℝ^{n} 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.

Original language | English (US) |
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Pages (from-to) | 1-23 |

Number of pages | 23 |

Journal | Proceedings of the London Mathematical Society |

Volume | 100 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2010 |

### ASJC Scopus subject areas

- Mathematics(all)