Let $\overline{M}=M \cup \partial M$ be a compact manifold with boundary, where $\partial M$ is the boundary of $\overline{M}$ and $M$ is the interior of $\overline{M}$. Let $P$ be an injective elliptic partial differential operator of order $2$ on $\overline{M}$.
For $k\geq 0$, let $u \in H^{2+k}$, the $L^2$-Sobolev space with $(2+k)$ weak derivatives. Assume that $Pu=f$ on $M$ and assume that $u|_{\partial M}=0$.
Question: Do we have (like in the case of compact manifolds without boundary, where we even have a slightly stronger estimate) an interior estimate of the form $$ ||u||_{H^{2+k}}^2 \leq C ( ||Pu||_{H^k}^2 + ||u||^2_{H^{1+k}})? $$
Taylor: Partial Differential Equations I - Basic Theory contains information about this that I don't understand. Equation (11.29) in Section 5 is what I want, but there is no statement that it holds in my case.
If $P$ has even degree and is strongly elliptic, which I'm happy to assume, then Proposition 11.10 in Section 5 states that the Dirichlet boundary condition is regular. Proposition 11.14 in Section 5 then gives an estimate in some complicated norms on a collar neighbourhood, but not the whole manifold with boundary.
Can you recommend another reference for this material?