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

Abstract

Let $1<p<\infty$ and let $\Omega$ be an open and bounded set of ${\mathbb{R}}^n$. We establish classical Korn inequalities \[ \inf_{\substack{v \in L^p(Ω)\\\mathcal A v = 0}} \|u - v\|_{W^{k,p}(Ω)} \le C \| \mathcal A u\|_{L^p(Ω)} \] 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-Müller projection estimate \[ \|u - Pu\|_{L^p(Ω)} \le C \| \mathcal A u\|_{W^{-k,p}(Ω)} \] 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.

41 pages; refurbished content

References in corpus (1)

• An elementary approach to the homological properties of constant-rank operators