Some three-dimensional problems related to dielectric breakdown and polycrystal plasticity

The well-known Sachs and Taylor bounds provide easy inner and outer estimates for the effective yield set of a polycrystal. It is natural to ask whether they can be improved. We examine this question for two model problems, involving 3D gradients and divergence-free vector fields. For 3D gradients the Taylor bound is far from optimal: we derive an improved estimate which scales differently when the yield set of the basic crystal is highly eccentric. For 3D divergence-free vector fields the Taylor bound may not be optimal, but it has the optimal scaling law. In both settings the Sachs bound is optimal.