Nonexistence of Solutions in Nonconvex Multidimensional Variational Problems

In the scalar n-dimensional situation, the extreme points in the set of certain gradient

-Young measures are studied. For n = 1, such Young measures must be composed from Diracs, while for n ≥ 2 there are non-Dirac extreme points among them, for n ≥ 3, some are even weakly* continuous. This is used to construct nontrivial examples of nonexistence of solutions of the minimization-type variational problem

\int_\Omega W(x,\nabla u)\,\mathrm{d}x ∫ Ω W ( x , ∇ u )   d x

with a Caratheodory (if n ≥ 2) or even continuous (if n ≥ 3) integrand W.