Differential inclusions and \(\mathcal A\)-quasiconvexity (Q1674188)

The paper considers problems of the form \[ v(x)\in E,\quad {\mathcal A}v=0, \] where \( {\mathcal A}\) is a first-order linear partial differential operator and the sets \(E\) are of the form \[ E=\{\xi \in {\mathbb R}^n;\quad F_i(\xi )=0,\quad i=1,\dots,N\}, \] where \(F_i:{\mathbb R}^n\to {\mathbb R}\), \(i=1,\dots,N\) are continuous and \({\mathcal A}\)-quasiconvex. The above problem is studied with constant boundary conditions. In the case \(N=1\), under certain hypotheses, it is established an abstract existence result. A similar existence result is obtained also for \(N>1\), but under more restrictive hypotheses.