A Variational Model for Dislocations in the Line Tension Limit

Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocations dynamics

“…Indeed, in many references in physics, the authors describe dislocation dynamics by line tension terms deriving from an energy associated to the dislocation line. See for instance Brown [18] and Barnet and Gavazza [14] for physical references, and Garroni and Müller [29], [28] for a variational approach. As far as we know, our result is the first rigorous proof for the convergence of dislocation dynamics to mean curvature motion.…”

“…The work of Garroni and Müller [28] suggests that we should have x,λ 1 2 (c ε 0 ρ ε λ )ρ ε λ → dλ λ G(Du 0 /|Du 0 |), where λ is the λ level set of u 0 . We deduce that (using formally the coarea formula for BV functions) and so, formally, E ε (u ε ) → E(u 0 ).…”

A nonlocal singular perturbation problem with periodic well potential

“…It is also possible to derive the Γ-convergence part of Theorem 1.1 from the result of [1] by a cutoff argument like in [3], but this approach does not lead to the compactness results obtained here. Proof.…”

“…We choose δ sufficiently small so πQ 1 > 4 3 (this also defines σ) and η 1 so small that η log µ η < 3 4 K for η < η 1 , so πQ 1 η log µ η > K. For s < 1 πQ1η , we can also estimate −πsQ 1 log(2σ) < − log(2σ). Using the definitions of T , L, and L 0 , we obtain that…”

On the behavior of dissipative systems in contact with a heat bath: Application to Andrade creep

“…This model has been proposed by Koslowski, Cuitiño and Ortiz in [19] and a mathematically rigorous analysis of the line-tension limit can be found in [15,16]. Secondly, this infinite-dimensional evolution will be reduced to a finite-dimensional one that takes account of the values of the phase field at certain "obstacle sites".…”