Journal Article10.1137/S0036141094263767
Invertibility and a Topological Property of Sobolev Maps
66
TL;DR: In this paper, the relationship between continuous and injective homeomorphisms is analyzed when the continuity and invertibility assumptions are weakened, and the relationship is shown to be nonlinear elasticity.
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Abstract: Let O be a bounded domain in Rn, let d : O¯ ? O¯ be a homeomorphism, and consider a function u : O¯ ? Rn that agrees with d on ?O. If u is continuous and injective then u(O) = d(O). Motivated by problems in nonlinear elasticity the relationship between u(O) and d(O) is analyzed when the continuity and invertibility assumptions are weakened. Specifically maps that are continuous on almost every line and maps that lie in the Sobolev space W1,p with n - 1 < p < n are considered.
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