A basic property and useful tool in the theory of Sobolev spaces is
the density of smooth compactly supported functions in the space
(i.e., the functions with
weak derivatives of orders
to
in
). On
Riemannian manifolds, it is well known that the same property remains valid under
suitable geometric assumptions. However, on a complete noncompact manifold it can
fail to be true in general, as we prove here. This settles an open problem raised for
instance by E. Hebey (Nonlinear analysis on manifolds: Sobolev spaces and
inequalities,Courant Lect. Notes Math. 5 (1999), 48–49).
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