  Let $1 < p < \infty$ and let $\Omega$ be an open and bounded set of $\mathbb
R^n$. We establish classical Korn inequalities
  \[
  \inf_{\substack{v \in L^p(\Omega)\\\mathcal A v = 0}} \|u -
v\|_{W^{k,p}(\Omega)} \le C \| \mathcal A u\|_{L^p(\Omega)}
  \] for all $k$th order operators $\mathcal A$ satisfying the maximal-rank
condition. This new condition is satisfied by the divergence, Laplacian,
Laplace-Beltrami, and Wirtinger operators, among others. As such, our estimates
generalize Fuchs' estimates for the del-bar operator to maximal-rank operators
and to arbitrary domains. For domains with sufficiently regular boundary
$\partial \Omega$, we are able to construct an $L^p(\Omega)$-bounded projection
$P$, onto the kernel of the operator. This projection is shown to satisfy a
classical Fonseca-M\"uller projection estimate
  \[
  \|u - Pu\|_{L^p(\Omega)} \le C \| \mathcal A u\|_{W^{-k,p}(\Omega)}
  \] as well as analogous estimates for higher-order derivatives. As a
particular application of our results, we are able to establish a weak Korn
inequality for general constant-rank operators (by taking the infimum over all
$\mathcal A$-harmonic maps instead of taking it over all $\mathcal A$-free
maps). Several examples are discussed.
