Polyconvexity and counterexamples to regularity in the calculus of variations

Using a technique explored in unpublished work of Ball and Mizel I shall

show that already in 2 and 3 dimensions there are vectorfields which are

singular minimizers of integral functionals whose integrand is strictly

polyconvex and depends on the gradient of the map only. The analysis behind

these results gives rise to an interesting question about the relationship

between the regularity of a polyconvex function and that of its possible

convex representatives. I shall indicate why this question is interesting in

the context of the regularity results above and I shall answer it in certain

cases.