sobolev spaces between manifolds
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2019 ◽  
Vol 12 (3) ◽  
pp. 303-332 ◽  
Author(s):  
Alexandra Convent ◽  
Jean Van Schaftingen

AbstractWe 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\geq 1} and {k\geq 2}, this allows us to define the intrinsic higher-order homogeneous Sobolev space {\dot{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 {\dot{W}^{k,p}(M,N)} is that {\pi_{\lfloor kp\rfloor}(N)\simeq\{0\}}. We investigate the chain rule for higher-order differentiability in this setting.





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