Existence of minimizers for non-quasiconvex integrals (Q1905662)

In this paper the authors give conditions for existence and non-existence of solutions of a problem of the type \[ \inf\Biggl\{F(u)= \int_\Omega f(Du(x))dx,\quad u\in u_0+ W^{1,\infty}_0(\Omega, \mathbb{R}^N)\Biggr\} \] with linear boundary data \(u_0\) and non-quasiconvex functions \(f\). Among others, their detailed analysis shows that there is more hope to solve the minimization problem in the vectorial case \((N> 1)\) than in the scalar case \((N= 1)\). People interested in these topics can find in this paper a rich bibliography and a variety of tools and examples.