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A note on the local invertibility of Sobolev functions

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  • Mathematics


omslaga 9..9 {orders}ms/990063/putten.3d -20.11.00 - 10:36 A NOTE ON THE LOCAL INVERTIBILITY OF SOBOLEV FUNCTIONS ROBERTO VAN DER PUTTEN Abstract. We give some topological and analytical conditions in order that a continuous Sobolev function be a local homeomorphism. The results are obtained in the setting of the spaces W 1;n ;Rn and W 2;p ;Rn . 1. Introduction. In this paper we deal with the local invertibility of continuous mappings and, more precisely, with the properties of the branch set of such mappings; we recall that, if is an open subset of Rn and f : ! Rn a continuous map- ping, the branch set of f , denoted by Bf , is the set of all points x 2 where f does not de¢ne a local homeomorphism. It is well known that if f 2 C1, then Bf � Zf where Zf fx 2 : Df x exists and detDf x 0g, but the study of Bf becomes more di⁄cult beyonds the class of smooth mappings. Some results have been obtained under topological assumptions: if f is light and sense-preserving (see below for de¢nitions) then the topological dimension of Bf and f Bf is not greater than nÿ 2 and Bf � Zf [ Sf 1:1 where Sf fx 2 : f is not weakly di¡erentiable at xg ([11] and [3]). However, it is not known under what analytical conditions a mapping is light and sense-preserving ; some results can be found in [7] (mappings with ¢nite dilatation) and in the monograph of Rickman ([11]) on quasiregular mappings. Invertibility has been studied also in the setting of nonlinear elasticity: in fact this requirement guarantees that interpenetration of matter does not occur. In this case Ball and Sï verak ([2], [13]) have found analytical condi- tions which implies the global invertibility of Sobolev functions. MATH. SCAND. 83 (1998), 255^264 Received March 11, 1996. {orders}ms/990063/putten.3d -20.11.00 - 10:37 In this paper we present three results in the setting of Sobolev spaces: the ¢rst two concern mappings belonging to W 1;n ;Rn and they are slight im- provements of the recalled r

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