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

TL;DR

This paper proves generalized Korn inequalities and constructs bounded projections for a broad class of constant-rank differential operators on domains, extending classical results to new operators and domain types.

Contribution

It introduces a maximal-rank condition for operators, generalizing Korn inequalities and projection estimates to a wider class of differential operators and domains.

Findings

Established Korn inequalities for maximal-rank operators.

Constructed bounded projections onto operator kernels.

Extended classical estimates to new operators and arbitrary domains.

Abstract

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