Convex integration: a method of approximate solutions (Q2455128)

The author introduces the concept of small successive perturbations for the sequence \(u_j\in W^{1,1}(\Omega,\mathbb{R}^m)\), and shows that every sequence \(u_j\) \((j= 1,2,\dots)\) obtained by small successive perturbations converges strongly in \(L^1(\Omega)\), with its gradients converging in measure, in the case when \(\| u_j\|_{W^{1,1}(\Omega)}\leq C\). Here \(\Omega\) is a bounded domain in \(\mathbb{R}^d\) and \(W^{1,1}(\Omega,\mathbb{R}^m)\) is a space of Sobolev functions \(u: \Omega\subset\mathbb{R}^d\to \mathbb{R}^m\). The convergence result mentioned above is applied to prove solvability and construct solutions to the particular class of differential inclusions.