Geodesic
The closure of planar diffeomorphisms in Sobolev spaces
Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 1, pp. 181-224

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We characterize the (sequentially) weak and strong closure of planar diffeomorphisms in the Sobolev topology and we show that they always coincide. We also provide some sufficient condition for a planar map to be approximable by diffeomorphisms in terms of the connectedness of its counter-images, in the spirit of Young's characterisation of monotone functions. We finally show that the closure of diffeomorphisms in the Sobolev topology is strictly contained in the class INV introduced by Müller and Spector.

DOI : 10.1016/j.anihpc.2019.08.001
Keywords: Sobolev approximation of mappings, INV mappings, Non-crossing mappings 
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     author = {De Philippis, G. and Pratelli, A.},
     title = {The closure of planar diffeomorphisms in {Sobolev} spaces},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {181--224},
     publisher = {Elsevier},
     volume = {37},
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     year = {2020},
     doi = {10.1016/j.anihpc.2019.08.001},
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     zbl = {1446.46018},
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De Philippis, G.; Pratelli, A. The closure of planar diffeomorphisms in Sobolev spaces. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 1, pp. 181-224. doi: 10.1016/j.anihpc.2019.08.001

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