Convex integration with constraints and applications to phase transitions and partial differential equations (Q1969507)

Let \(\Omega\subset \mathbb{R}^n\) be bounded and let \(u:\Omega\to \mathbb{R}^m\) be a Lipschitz map. Also consider a subset \(K\) in the space of \(m\times n\) matrices. The authors study the first-order partial differential relation \(Du\in K\), a.e. in \(\Omega\). Motivated by questions of crystal microstructure they extend Gromov's theory of convex integration in two ways: first, by allowing additional constraints on the minors of \(Du\) and second, by replacing Gromov's \(P\)-convex hull (in the context of this paper ``\(P\)-convexity'' is also equivalent with ``laminated convexity'') by the rank-one convex hull. (Rank-one convex hulls have been used recently by the authors to construct nowhere regular solutions to variational elliptic systems).