Let $ (M,g) $ be a Riemannian manifold ($ g$ Riemannian metric) and let $ f: M \rightarrow R $ be a Lipschitz function (with respect to $ g $) with compact support. I want to study if it is possible to approximate $ f $ with smooth functions with compact support with respect to the norm of the Sobolev space $ H^p_1 $.
I know an argument that solves the problem when $ M $ is complete. The argument is the following:
Since $ f $ is a Lipschitz function with compact support then $ f \in H^p_1(M) $ (for every $ p \geq 1 $). Now, since $ M $ is complete the set of smooth functions with compact support is dense in $ H^p_1(M) $.
Now my question:
Is it possible to extend this result to non complete case? In detail if $ M $ is not complete, is it possible to approximate a lipschitz function with compact support (with respect to the norm of $ H^p_1 $) with smooth functions ($ C^\infty $) with compact support ?