Quantitative characterization of traces of Sobolev maps

Mazowiecka, Katarzyna Ewa;Van Schaftingen, Jean
(2023) Communications in Contemporary Mathematics — Vol. 25, n° 02, p. 2250003 (2023)

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We give a quantitative characterization of traces on the boundary of Sobolev maps in Ẇ1,p(M,N), where M and N are compact Riemannian manifolds, ∂M≠∅: the Borel-measurable maps u:∂M→N that are the trace of a map U∈Ẇ1,p(M,N) are characterized as the maps for which there exists an extension energy density w:∂M→[0,∞] that controls the Sobolev energy of extensions from ⌊p−1⌋-dimensional subsets of ∂M to ⌊p⌋-dimensional subsets of M.
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Mazowiecka, K. E., & Van Schaftingen, J. (2023). Quantitative characterization of traces of Sobolev maps. Communications in Contemporary Mathematics, 25(02), 2250003. https://doi.org/10.1142/s0219199722500031 (Original work published 2023)