Cauchy-Dirichlet problem for first order nonlinear systems (Q1381531)

The problem of finding \(u\in W^{1,\infty}(\Omega;\mathbb{R}^m)\) satisfying \(\nabla u\in E\) a.e. on a domain \(\Omega\subset\mathbb{R}^n\) and \(u=\varphi\) on the boundary \(\partial\Omega\) is addressed. For \(E\subset\mathbb{R}^{m\times n}\) compact, having the so-called polyconvex hull equal to the so-called rank-one-convex hull, and satisfying still so-called segment and extreme-point properties, this problem is shown to have infinitely many solutions provided \(\varphi\in C^1(\overline\Omega;\mathbb{R}^m)\) has the gradient \(\nabla\varphi\) in \(E\) or in the interior of the polyconvex hull of \(E\). These solutions are even dense in \(W^{1,\infty}(\Omega;\mathbb{R}^m)\). Various applications and generalizations (e.g. \(x\)- and \(u\)-dependence of \(E)\) are mentioned, too.