Let's consider an elliptic (vectorial, homogeneous, constant coefficient) system of order 2m $$ \begin{cases} Lu=f &\text{in }\Omega\\ Bu=0 &\text{on }\partial\Omega \end{cases} $$ which is self adjoint, meaning that $$ \int_{\Omega}\langle Lu,v \rangle \mathrm{d}x=\int_{\Omega}\langle u,Lv\rangle\mathrm{d} x $$ for all smooth $u,\,v$ that satisfy the boundary condition $Bu=0=Bv$.
My question is: where could I find satisfactory $L^2$-theory for such an elliptic boundary value problem, by which I mean that for each $f\in H^{-m}(\Omega)$ there exists a unique solution $u\in H^m(\Omega)$ with the estimate $$ \|u\|_{H^m(\Omega)}\leq C\|f\|_{H^{-m}(\Omega)}? $$
Equivalently, does anyone know whether the Lopatinskii--Shapiro conditions are satisfied for fully self adjoint elliptic problems?
(Almost) equivalently, does anyone know whether the a priori estimate $$ \|u\|_{H^m(\Omega)}\leq C\|Lu\|_{H^{-m}(\Omega)} $$ holds for smooth fields satisfying $Bu=0$?
Other details/assumptions: I am happy to assume that $\Omega\subset\mathbb R^n$ is a bounded and smooth domain. By elliptic operator/system above I mean $$ (Lu)_j=\sum_{|\alpha|=2m} \langle L_{j}^\alpha, \partial^\alpha u\rangle,\quad u\colon\Omega\rightarrow\mathbb R^N $$ for $L_j^\alpha\in\mathbb R^{N}$, $j=1\ldots N$, which is such that the (symmetric) matrix defined by $$ L(\xi)_j=\sum_{|\alpha|=2m} L_{j}^\alpha \xi^\alpha\in\mathbb R^{N},\quad \xi\in\mathbb R^n $$ is strictly positive definite for all $\xi\in\mathbb R^n\setminus\{0\}$. Equivalently, $$ \langle u_0, L(\xi)u_0\rangle\geq C|\xi|^{2m}|u_0|^2,\quad u_0\in\mathbb R^N,\,\xi\in\mathbb R^n, $$ or equivalently $$ \det L(\xi)\neq0,\quad\text{for all }\xi\in\mathbb R^n\setminus\{0\}. $$ Finally, I can add more assumptions on $L$, so please let me know if you know more restrictive statements.
Thank you!