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.
@article{AIHPC_2020__37_1_181_0, 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}, number = {1}, year = {2020}, doi = {10.1016/j.anihpc.2019.08.001}, mrnumber = {4049920}, zbl = {1446.46018}, language = {en}, url = {https://www.numdam.org/articles/10.1016/j.anihpc.2019.08.001/} }
TY - JOUR AU - De Philippis, G. AU - Pratelli, A. TI - The closure of planar diffeomorphisms in Sobolev spaces JO - Annales de l'I.H.P. Analyse non linéaire PY - 2020 SP - 181 EP - 224 VL - 37 IS - 1 PB - Elsevier UR - https://www.numdam.org/articles/10.1016/j.anihpc.2019.08.001/ DO - 10.1016/j.anihpc.2019.08.001 LA - en ID - AIHPC_2020__37_1_181_0 ER -
%0 Journal Article %A De Philippis, G. %A Pratelli, A. %T The closure of planar diffeomorphisms in Sobolev spaces %J Annales de l'I.H.P. Analyse non linéaire %D 2020 %P 181-224 %V 37 %N 1 %I Elsevier %U https://www.numdam.org/articles/10.1016/j.anihpc.2019.08.001/ %R 10.1016/j.anihpc.2019.08.001 %G en %F AIHPC_2020__37_1_181_0
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. https://www.numdam.org/articles/10.1016/j.anihpc.2019.08.001/
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