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

Abstract

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 three–dimensional gradients and divergence–free vector fields. For three–dimensional gradients, the Taylor bound is far from optimal: we derive an improved estimate that scales differently when the yield set of the basic crystal is highly eccentric. For three–dimensional 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.