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

Abstract

We prove that if A : D(A) ⊂ L p (Ω; V ) → L p (Ω; W ) is a k th order constant-rank differential operator with maximal rank, then there exists a linear solution operator A −1 satisfying Sobolev regularity estimates. This allows us to construct a linear projection T :for all sufficiently regular maps. The estimate generalizes Fuch's distance estimate for the del-var operator, to operators such as the laplacian and the divergence. On regular domains, the same estimates are (trivially) observed for operators with finite dimensional null-space. When A is only assumed to be of constant rank, we are able to show that the Sobolev distance to the space of A-harmonic maps is bounded by Au, that is, minSimilar projection and distance estimates involving lower-order derivatives are further discussed, as well as a plethora of examples and applications with well-known operators. Contents 1. Introduction and main results 1 2. Set-up and preliminary results 10 3. Proofs of the main results 15 4. Examples of maximal-rank operators and applications 19 References 23

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

“…This directly follows from Lemma 4.1. Indeed, applying Lemma 4.1 k-times, there exists a finite dimensional space V of polynomials such that (2) , the result directly follows. Finally, (c) is immediate by applying (b) in both directions.…”

“…Corollary 4.2 (Kernels of annihilators). Let A (1) and A (2) be two homogeneous differential operators of order k (1) and k (2) , which have constant rank over C and both act on C ∞ (R n ; R d ). Moreover, suppose that their Fourier symbols satisfy ker(A (1) [ξ]) ⊂ ker(A (2) [ξ]) for all ξ ∈ C n .…”

“…where V is a finite dimensional vector space (consisting of polynomials). (c) If, in addition, ker(A (1) [ξ]) = ker(A (2) [ξ]), then we may write…”

“…We aim to apply Theorem 3.2, and we explain the setting first. Assuming that A (1) is R l1 -valued and A (2) is R l2 -valued, we may write for…”

“…Remark 5.8. To conclude, let us remark that another approach to the problem described in this section is discussed in [2,Lem. 14] for operators of maximal rank.…”