Calculus of Variations and Geometric Measure Theory
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A. Monteil - J. Van Schaftingen

Uniform boundedness principles for Sobolev maps into manifolds

created by monteil on 26 Sep 2017



Inserted: 26 sep 2017

Year: 2017

ArXiv: 1709.08565 PDF


Given a connected Riemannian manifold $\mathcal{N}$, an \(m\)--dimensional Riemannian manifold $\mathcal{M}$ 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 $W^{s,p}(\mathcal{M}, \mathcal{N})$ imply corresponding uniform quantitative bounds. This result is a nonlinear counterpart of the classical Banach--Steinhaus uniform boundedness principle in linear Banach spaces.

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