JournalsaihpcVol. 36, No. 2pp. 417–449

Uniform boundedness principles for Sobolev maps into manifolds

  • Antonin Monteil

    Université Catholique de Louvain, Institut de Recherche en Mathématique et Physique, Chemin du Cyclotron 2 bte L7.01.01, 1348 Louvain-la-Neuve, Belgium
  • Jean Van Schaftingen

    Université Catholique de Louvain, Institut de Recherche en Mathématique et Physique, Chemin du Cyclotron 2 bte L7.01.01, 1348 Louvain-la-Neuve, Belgium
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Abstract

Given a connected Riemannian manifold N\mathscr{N}, an mm-dimensional Riemannian manifold M\mathscr{M} which is either compact or the Euclidean space, p[1,+)p \in [1, + \infty) and s(0,1]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 Ws,p(M,N)W^{s,p}(\mathscr{M},\mathscr{N}) imply corresponding uniform quantitative bounds. This result is a nonlinear counterpart of the classical Banach–Steinhaus uniform boundedness principle in linear Banach spaces.

Cite this article

Antonin Monteil, Jean Van Schaftingen, Uniform boundedness principles for Sobolev maps into manifolds. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 2, pp. 417–449

DOI 10.1016/J.ANIHPC.2018.06.002