Regularity of higher order elliptic regularity on compact smooth manifolds with boundary

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 $partial M$, $m in mathbb{N}$, $$L = sum_{|alpha| leqslant m, ,|beta| leqslant m} partial^{alpha}(a_{alphabeta} partial^beta)$$ 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!