Uniform boundedness principles for Sobolev maps into manifolds

Monteil, Antonin; Van Schaftingen, Jean

doi:10.1016/j.anihpc.2018.06.002

Monteil, Antonin ; Van Schaftingen, Jean

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 2, pp. 417-449.

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Résumé

Given a connected Riemannian manifold $N$ , an m-dimensional Riemannian manifold $M$ which is either compact or the Euclidean space, $p \in [1, + \infty)$ and $s \in (0, 1]$ , we establish, for the problems of surjectivity of the trace, of weak-bounded approximation, of lifting and of superposition, that qualitative properties satisfied by every map in a nonlinear Sobolev space $W^{s, p} (M, N)$ imply corresponding uniform quantitative bounds. This result is a nonlinear counterpart of the classical Banach–Steinhaus uniform boundedness principle in linear Banach spaces.

DOI : 10.1016/j.anihpc.2018.06.002

Classification : 46T20, 46E35, 46T10, 58D15
Mots-clés : Nonlinear uniform boundedness principle, Opening of maps, Counterexamples

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     title = {Uniform boundedness principles for {Sobolev} maps into manifolds},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2018.06.002/}
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Monteil, Antonin; Van Schaftingen, Jean. Uniform boundedness principles for Sobolev maps into manifolds. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 2, pp. 417-449. doi : 10.1016/j.anihpc.2018.06.002. http://www.numdam.org/articles/10.1016/j.anihpc.2018.06.002/

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