I have trouble in finding a source in the literature for the following result:
Let $\overline{M}$ be a compact smooth manifold of dimension $n \in \mathbb{N}$ with interior $M$ and non-empty boundary $\partial M$, $m \in \mathbb{N}$, $$L = \sum_{|\alpha| \leqslant m, \,|\beta| \leqslant m} \partial^{\alpha}(a_{\alpha\beta} \partial^\beta)$$ be a differential operator of order $2m$ with $a_{\alpha \beta } \in C^{\min \{|\alpha|,|\beta| \}}(\overline{M};\mathbb{R}) \,\,(\alpha, \beta \in \mathbb{N}^n,\,|\alpha| \leqslant m,\, |\beta|\leqslant m)$ that is uniformly elliptic, i.e. \begin{align*} \sum_{|\alpha| = m,\,|\beta| = m} a_{\alpha,\beta} \xi^{\alpha + \beta} \geqslant c|\xi|^{2m} \,\,\,\,\,(\xi \in \mathbb{R}^n) \end{align*} for a $c > 0$. Let further $f \in L_2(M;\mathbb{R})$ and $u \in H^m(M;\mathbb{R})$ with $$Lu = f.$$ Then $u \in H^{2m}(M;\mathbb{R})$.
Because I don't exactly know if this result is stated correctly, I apologize for any mistakes in the formulation of this result. Nevertheless, can eventually someone give me some tip where to find this result in the literature?
Thank you in advance!
The result is certainly true and is classical, though the details are rarely written out in full. One textbook where it can be found however is in Vol. 1, Ch. 5, Sec. 11 of
Taylor, Michael E., Partial differential equations. I: Basic theory, Applied Mathematical Sciences 115. New York, NY: Springer (ISBN 978-1-4419-7054-1/hbk; 978-1-4419-7055-8/ebook). xxii, 654 p. (2011). ZBL1206.35002.
The idea is not too different from the second order case; working locally we can reduce to the case of the full and half space, use ellipticity to establish a coercivity estimate, and bootstrap using difference quotients. Note that estimating the normal derivative is more complicated here, and while the idea of using the equation remains the same, one needs to work in negative order spaces to infer regularity of $\partial_n^{2m}u.$
There is another more general approach of Agmon, Douglis, & Nirenberg (1959) involving constructing explicit potentials, but that's not any easier than the above.