Higher order intrinsic weak differentiability and Sobolev spaces between manifolds
(2019) Advances in Calculus of Variations — Vol. 12, n° 3, p. 303-332 (2019)
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- Abstract
- We define the notion of higher-order colocally weakly differentiable maps from a manifold M to a manifold N. When M and N are endowed with Riemannian metrics, p ≥ 1 and k ≥ 2, this allows us to define the intrinsic higher-order homogeneous Sobolev space W k , p ≥ ( M , N ). We show that this new intrinsic definition is not equivalent in general with the definition by an isometric embedding of N in a Euclidean space; if the manifolds M and N are compact, the intrinsic space is a larger space than the one obtained by embedding. We show that a necessary condition for the density of smooth maps in the intrinsic space W k,p(M, N) is that π [kp](N) E {0}. We investigate the chain rule for higher-order differentiability in this setting.
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Convent, A., & Van Schaftingen, J. (2019). Higher order intrinsic weak differentiability and Sobolev spaces between manifolds. Advances in Calculus of Variations, 12(3), 303-332. https://doi.org/10.1515/acv-2017-0008 (Original work published 2019)