A generalized Poincaré inequality for a class of constant coefficient differential operators

Abstract

Abstract. We study first order differential operators P = P(D) with constant coefficients. The main question is under what conditions the following full gradient L p estimate holds:We show that the constant rank condition is sufficient. The concept of the Moore-Penrose generalized inverse of a matrix comes into play.

Homogenization of integral energies under periodically oscillating differential constraints

“…For future use, we remark that the conclusion remains valid when φ ∈ S (R d ; R N ), as the Fourier analysis on which the proof is based is still viable in this case, or even when φ ∈ W k,p (R d ; R N ), by approximation. The sufficiency of the constant rank condition had been already observed in the literature, see for instance the bibliography in [50] or the paper by D. Gustafson [52], where a proof for first order operators is given.…”

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

“…First, we address the existence and properties of A −1 on L p (Ω; W ). The strategy of the proof is to exploit the regularity properties of the Moore-Penrose pseudo-inverse of the principal symbol of A, a technique that dates back to the work of Gustafson [11] and more recently also key for the work of Raita [16]. Let us write k = N + ℓ > 0, where ℓ is an integer, so that ℓ > −N .…”

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

“…1 The use of multipliers defined by the Moore-Penrose pseudo-inverse of the principal symbol as a means to establish Poincaré-type estimates for constant-rank operators dates back to the work of Gustafson [Gus11].…”

“…The first assertion follows by setting u = F −1 (B † F v) as in the previous proof. The gradient estimate follows from Gustafson's strategy [Gus11] of using the pseudo-inverse of the symbol to as a means to establish Calderón-Zygmund estimates: derivating this identity for u (k ′ times) and appealing to the properties of the Fourier transform we find that (recall that B † is smooth and homogeneous of degree −k ′ away from zero) D k ′ u = F −1 (T F v) where T is smooth and homogenenous of degree zero on R d − {0}. Thus, by the L p -theory of singular integrals , there exists a constant C(A, p) (d and B depend on A) such that…”