Intrinsic colocal weak derivatives and Sobolev spaces between manifolds

Convent, Alexandra;Van Schaftingen, Jean
(2016) Scuola Normale Superiore di Pisa. Annali. Classe di Scienze — Vol. 16, n° 1, p. 97-128 (2016)

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Abstract
We define the notion of colocally weakly differentiable maps from a manifold M to a manifold N. If p≥1 and M and N are endowed with a Riemannian metric, this allows us to define intrinsically the homogeneous Sobolev space W˙1,p(M,N). This new definition is equivalent with the definition by embedding in the Euclidean space and with the definition of Sobolev maps into a metric space. The colocal weak derivative is an approximate derivative. The colocal weak differentiability is stable under the suitable weak convergence. The Sobolev spaces can be endowed with various intrinsinc distances that induce the same topology and for which the space is complete.
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Convent, A., & Van Schaftingen, J. (2016). Intrinsic colocal weak derivatives and Sobolev spaces between manifolds. Scuola Normale Superiore di Pisa. Annali. Classe di Scienze, 16(1), 97-128. https://doi.org/10.2422/2036-2145.201312_005 (Original work published 2016)