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Let $1 < p < \infty$ and let $\Omega$ be an open and bounded set of $\mathbbR^n$. 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-rankcondition. This new condition is satisfied by the divergence, Laplacian,Laplace-Beltrami, and Wirtinger operators, among others. As such, our estimatesgeneralize Fuchs' estimates for the del-bar operator to maximal-rank operatorsand 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 aclassical 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 aparticular application of our results, we are able to establish a weak Korninequality for general constant-rank operators (by taking the infimum over all$\mathcal A$-harmonic maps instead of taking it over all $\mathcal A$-freemaps). Several examples are discussed.

New projection and Korn estimates for a class of constant-rank operators on domains