The Failure of Rank-One Connections

Abstract

This article is concerned with interface problems for Lipschitz mappings f ₊:ℝⁿ ₊→ℝⁿ and f ₋:ℝⁿ ₋→ℝⁿ in the half spaces, which agree on the common boundary ℝ^{n − 1}=∂ℝⁿ ₊=∂ℝⁿ ₋. These naturally occur in mathematical models for material microstructures and crystals. The task is to determine the relationship between the sets of values of the differentials Df ₊ and Df ₋. For some time it has been thought that the polyconvex hulls [Df ₊]^pc and [Df ₋]^pc satisfy Hadamard's jump condition or are at least rank-one connected. Our examples here refute this idea.

The estimates of the Jacobians we obtain in the course of solving the so-called Monge-Ampère inequalities seem also to be of independent interest. As an application, we construct uniformly elliptic systems of first order partial differential equations in the same homotopy class as the familiar Cauchy-Riemann equations, for which the unique continuation property fails.