Uniform boundedness principles for Sobolev maps into manifolds

2017/09/25 by Antonin Monteil, Jean Van Schaftingen
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Equations and Boundary Problems #Mathematical analysis #Mathematics #Nonlinear Partial Differential Equations #Pure mathematics #Sobolev space #math.AP #math.FA #msc:46T10 #msc:46T20

paper · pdf · doi:10.1016/j.anihpc.2018.06.002

published as Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), n. 2, 417-449 · 28 pages

arxiv created 2017/09/25 · openalex publication_date 2018/06/20 · arxiv updated 2019/04/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Given a connected Riemannian manifold N , an m -dimensional Riemannian manifold M which is either compact or the Euclidean space, p ∈ [1, + ∞) and s ∈ (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.

Citations

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• Boundary regularity and the Dirichlet problem for harmonic maps
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