Page 483 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 483
GEOMETRIC-FUNCTIONAL INEQUALITIES AND RELATED TOPICS (MS-23)
Reverse superposition estimates, lifting over a compact covering and
extensions of traces for fractional Sobolev mappings
Jean Van Schaftingen, Jean.VanSchaftingen@uclouvain.be
UCLouvain, Belgium
Coauthor: Petru Mironescu
When u ∈ W 1,p(Rm), then |u| ∈ W 1,p(Rm) and
|Du|p = |D|u||p;
Rm Rm
this provides an a priori control on u by |u| in first-order Sobolev spaces. For fractional Sobolev
spaces when sp > 1, we prove a reverse oscillation inequality that yields a control on u by |u|
in W s,p(Rm). As another consequence of the reverse oscillation estimate, given a covering map
π : N → N , with N compact, we prove any u ∈ W s,p(Rm, N ) has a lifting, that is, can be
written as u = π ◦ u, with u ∈ W s,p(Rm, N ). This completes the picture for lifting of fractional
Sobolev maps and implies the surjectivity of the trace operator on Sobolev spaces of mappings
into a manifold when the fundamental group is finite.
481
Reverse superposition estimates, lifting over a compact covering and
extensions of traces for fractional Sobolev mappings
Jean Van Schaftingen, Jean.VanSchaftingen@uclouvain.be
UCLouvain, Belgium
Coauthor: Petru Mironescu
When u ∈ W 1,p(Rm), then |u| ∈ W 1,p(Rm) and
|Du|p = |D|u||p;
Rm Rm
this provides an a priori control on u by |u| in first-order Sobolev spaces. For fractional Sobolev
spaces when sp > 1, we prove a reverse oscillation inequality that yields a control on u by |u|
in W s,p(Rm). As another consequence of the reverse oscillation estimate, given a covering map
π : N → N , with N compact, we prove any u ∈ W s,p(Rm, N ) has a lifting, that is, can be
written as u = π ◦ u, with u ∈ W s,p(Rm, N ). This completes the picture for lifting of fractional
Sobolev maps and implies the surjectivity of the trace operator on Sobolev spaces of mappings
into a manifold when the fundamental group is finite.
481
