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

Abstract

Using properties of the alternating algebra, we give a surprisingly simple and constructive proof of Rait , ȃ's result that every constant-rank operator A possesses an exact potential B and an exact annihilator Q. Our construction is completely self-contained and improves on the order of both the operators constructed by Rait , ȃ and the explicit formula for annihilators of elliptic operators due to Van Schaftingen. We also prove a generalized Poincaré lemma for a class of mean-value zero Schwartz functions defined on full-space.

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

“…In this vein, Schulenberger and Wilcox [17,18] introduced the constant-rank condition (5) ∀ξ ∈ R N − {0}, rank A(ξ) = const., which they proved to be a sufficient condition for the validity of the fullspace coercive estimate (4) when p = 2 (in which case (3) is equivalent to (4)). Inspired by the estimates of Schulenberger and Wilcox, Kato [13] and Murat [14] proved that the constant-rank property further implies that the canonical linear L 2 -projection T 2 : L 2 (Ω; V ) → N 2 (A, Ω) extends to an L p -bounded projection in the sense that…”

“…To work out the case when A is not a maximal-rank operator, we require to briefly discuss the dichotomy between constant-rank operators and elliptic complexes. Recently, Raita [16] (see also the simplified proof given in [5]) showed that there exists a surjective correspondence between elliptic complexes and constant-rank operators. There, the author proved that if A satisfies the constant rank property, then there exists a differential complex…”

Natural annihilators and operators of constant rank over $\mathbb{C}$

“…Following the works of Wilcox & Schulenberger [20] and Murat [17] (also see Fonseca & Müller [9]), operators A or B of the form (1.3) are said to be of constant rank (over R) provided dim R (A[ξ](V )) or dim R (B[ξ](W )), respectively, are independent of the phase space variable ξ ∈ R n \{0}. By Raita [18] (also see [3]), every constant rank operator possesses a constant rank potential.…”