Functions with prescribed singular values of the gradient (Q1265574)

The problem of finding \(u\in W^{1,\infty}(\Omega; \mathbb{R}^N)\) satisfying \(\nabla u\in K\) a.e. on a bounded domain \(\Omega\subset \mathbb{R}^N\) and \(u= u_0\) on the boundary \(\partial\Omega\) is addressed. For \(K(x)\) the set of matrices \(A\in \mathbb{R}^{N\times N}\) such that the eigenvalues of \(A^TA\) coincide with prescribed \(x\)-dependent values and for suitable boundary datum \(u_0\), this so-called ``singular-value'' problem is shown to have infinitely many solutions. The proof relies on Gromov's convex integration method.