Nonexistence of solutions of nonconvex multidimensional variational problems (Q2718680)

The authors study the nonexistence of solutions for problems of the form NEWLINE\[NEWLINE\text{Minimize }\int_\Omega W(x, \nabla u) dxNEWLINE\]NEWLINE in the class of functions \(u\in W^{1,p}(\Omega)\) such that \(\int_\Omega u = 0\), where \(\Omega\) is a bounded simply connected domain of \(R^n\), and \(W\) is a Carathéodory function satisfying a growth condition of the form \(c|s|^p \leq W(x,s)\leq C(1+|s|^p)\). They study the extreme points of the gradient \(L^p\)-Young measures. In the case \(n\geq 2\), they give examples of extreme points which are not Dirac measures. Using these extreme points, they construct problems which have no solution. In the case \(n=1\) extreme points are Dirac masses, and there always exist solutions to the above problem.