Dielectric breakdown: optimal bounds

Abstract

We study dielectric breakdown for composites made of two isotropic phases. We show that Sachs's bound is optimal. This simple example is used to illustrate a variational principle which departs from the traditional one. We also derive the usual variational principle by elementary means without appealing to the technology of convex duality

On the asymptotic behavior of variable exponent power–law functionals and applications

“…Similar problems can be raised in the analysis of models of dielectric breakdown and electrical resistivity, where an effective yield (strength) set is defined in essentially the same way, except that in these cases the pointwise constraint must be slightly modified, and (5) is replaced by the requirement that the field σ : → R 3 is either curl-free or divergence-free, respectively (see, e.g., [16,17], and [2]). In situations where a complete description of the effective yield set is not practically possible, one usually resorts to studying some natural inner and outer bounds for this set (the so-called Sachs and Bishop-Hill-Taylor bounds; see [17][18][19][20] for more details on these issues).…”

“…In situations where a complete description of the effective yield set is not practically possible, one usually resorts to studying some natural inner and outer bounds for this set (the so-called Sachs and Bishop-Hill-Taylor bounds; see [17][18][19][20] for more details on these issues). To the best of our knowledge, [16] is the first work where this type of problems is addressed by means of -convergence, leading to an efficient mathematical derivation of first-failure dielectric breakdown models (including the traditional one) as limiting cases of various power-law models, together with a variational characterization of the effective yield set in that setting by means of variational principles associated to the limiting functionals. More recently, Bocea and Nesi [2] have generalized the -convergence results in [16] to the A-quasiconvexity setting, allowing for more general linear PDE constraints on the underlying fields.…”

Quasiminima of the Lipschitz extension problem

“…[1,7,14,19], and partly because of that they have recently attracted considerable attention (see, for example, [5,6,8,20], and references therein). For simplicity, we restrict our attention to the archetypal model case F(x, u, Du) = |Du|, in case of which we already encounter the basic features of the theory but avoid some of the technicalities that could draw the attention away from the main ideas.…”

A Formal Derivation of the Aronsson Equations for Symmetrized Gradients

“…This is motivated in part by applications to problems arising in materials science (see, e.g. Garroni, Nesi, and Ponsiglione [13], Bocea and Nesi [9], and references therein). It turns out that minimizers u p of I p subject to boundary conditions converge, as p → ∞, to minimizers of the limiting functional…”