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

Abstract

We give a very concise proof of Ornstein's $L^1$ non-inequality for first- and second-order operators in two dimensions. The proof just needs a two-dimensional laminate supported on three points.

arXiv commentv2: small corrections and change to title

References in corpus (4)

• On Rank-One Convex Functions that are homogeneous of Degree One
• On the necessity of the constant rank condition for $L^p$ estimates
• Oscillation and concentration in sequences of PDE constrained measures
• Anisotropic Ornstein non inequalities

Cited by in corpus (5)

• Quasiconvexity, null Lagrangians, and Hardy space integrability under constant rank constraints
• Optimal incompatible Korn-Maxwell-Sobolev inequalities in all dimensions
• On the necessity of the constant rank condition for $L^p$ estimates
• The Dirichlet problem for the Jacobian equation in critical and supercritical Sobolev spaces
• Boundary ellipticity and limiting $L^1$-estimates on halfspaces