Deformations with finitely many gradients and stability of quasiconvex hulls (Q2712620)

The paper is concerned with solutions of the differential inclusion NEWLINE\[NEWLINE \nabla f\in K\quad\text{a.e. in }\Omega,\qquad f(x)=C(x)\quad \text{on }\partial\OmegaNEWLINE\]NEWLINE (with \(C\) given \(n\times m\) matrix) and with the related question of the stability of the quasiconvex hull \(K^{qc}\) of \(K\), defined as the set of all \(n\times m\) matrices \(C\) such that NEWLINE\[NEWLINE \inf\left\{\int_\Omega \text{dist}(K,\nabla f(x)) dx: f\in W^{1,\infty}(\Omega,{\mathbb R}^m),\;f(x)=C(x)\text{ on }\partial\Omega\right\}=0 NEWLINE\]NEWLINE (here \(\Omega\subset{\mathbb R}^n\) is an auxiliary open set). The results in the paper answer positively to a conjecture of Ball and James, about the existence of nontrivial Lipschitz maps with finitely many gradients without rank-one connections, and provide a large class of examples of sets \(K\) where the map \(K\mapsto K^{qc}\) is continuous, with respect to Hausdorff convergence. The approach to both problems is based on an ingenious and very flexible argument for solving differential inclusions, which somehow reconciles the Baire category approach and the convex integration approach, both used in the past by several authors to obtain existence results for the differential inclusion. The basic idea is that the map \(f\mapsto\nabla f\) is continuous with respect to the \(L^\infty\) topology on the domain NEWLINE\[NEWLINE{\mathcal B}:=\left\{f\in W^{1,\infty}(\Omega,{\mathbb R}^m):\nabla f\in U\text{ a.e.}\right\} NEWLINE\]NEWLINE and the strong topology \(L^1(\Omega,U)\) on the image for most (in the sense of Baire category) functions \(f\). For any such function \(f\) the gradient \(\nabla f\) has to stay in the parts of \(U\) without rank-one connections. The nice feature of this approach is that existence and stability come together.