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

Let $1<p<\mathrm{\infty }$ and let $\mathrm{\Omega }$ be an open and bounded set of $\mathbb R^n$. We establish classical Korn inequalities $$\underset{\begin{array}{c}v\in L^p(\mathrm{\Omega })\\ \mathcal{A}v=0\end{array}}{\inf }\mathrm{\parallel }u-v{\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}\le C\mathrm{\parallel }\mathcal{A}u{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}$$ 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 $\mathrm{\partial }\mathrm{\Omega }$, we are able to construct an $L^p(\mathrm{\Omega })$-bounded projection $P$, onto the kernel of the operator. This projection is shown to satisfy a classical Fonseca-M"uller projection estimate $$\mathrm{\parallel }u-Pu{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}\le C\mathrm{\parallel }\mathcal{A}u{\mathrm{\parallel }}_{W^{-k,p}(\mathrm{\Omega })}$$ 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.