On the necessity of the constant rank condition for $L^p$ estimates

Abstract

We consider a generalization of the elliptic $L^p$-estimate suited for linear operators with non-trivial kernels. A classical result of Schulenberger and Wilcox (Ann. Mat. Pura Appl. (4) 88: 229-305, 1971) shows that if the operator has constant rank then the estimate holds. We prove necessity of the constant rank condition for such an estimate.

arXiv comment5 pages; weaker versions of the results here had previously appeared in arXiv:1909.03923

References in corpus (1)

• Remarks on Ornstein's non-inequality in ${\mathbb{R}}^{2\times 2}$

Cited by in corpus (6)

• Quasiconvexity, null Lagrangians, and Hardy space integrability under constant rank constraints
• Oscillation and concentration in sequences of PDE constrained measures
• Compensated compactness: continuity in optimal weak topologies
• Remarks on Ornstein's non-inequality in ${\mathbb{R}}^{2\times 2}$
• A variational view on constitutive laws in parabolic problems
• Syzygies, constant rank, and beyond