Let $L$ be a differential operator of order $k$ defined on a bounded open subset $U$ of a Riemannian manifold $M$ that is elliptic when restricted to a smaller open subset $V$ such that $\bar V\subseteq U$.
Question 1: Is $L$ a Fredholm operator on $C^\infty(V)$? (Or maybe it's more appropriate to consider it as an operator $H^k(V)\to L^2(V)$?)
Question 2: Does there always exist an elliptic differential operator $\bar L$ on $M$ of the same order $k$ that extends $L$?
Thoughts:
For Q1, I know that an elliptic operator on a closed manifold has a parametrix and so is Fredholm. Intuitively I would think this works for bounded open domains as well, but I'm not sure. It would be nice to find a reference for this.
For Q2, perhaps one can try to extend the symbol of $L$ to all of $M$ so that it is invertible outside the zero section of $T^*M$, but I'm not sure how to do this.