A characterization of families of function sets described by constraints on the gradient (Q1904221)

The authors give some characterization of the families of subsets \(K(G)\) of \(W^{1,p}(\mathbb{R}^n)\), for example of the form \[ K(G)= \{u\in W^{1,p}(\mathbb{R}^n): Du(x)\in C\text{ a.e. in }G\}, \] where \(C\) is a closed convex subset of \(\mathbb{R}^n\). Moreover, they study the characterization of families of subsets of BV and \(L^p\). The main tools for the proofs are a ``blow-up'' argument and an approximation procedure of continuously differentiable functions by piecewise affine functions. Some applications to the study of the homogenization of elastic-plastic torsion of a cylindrical bar are also given.