Rectifiability of entropy defect measures in a micromagnetics model

Abstract

We study the fine properties of a class of weak solutions u of the eikonal equation arising as asymptotic domain of a family of energy functionals introduced in (Rivière T, Serfaty S. Limiting domain wall energy for a problem related to micromagnetics. Comm Pure Appl Math 2001; 54(3):294-338). In particular we prove that the entropy defect measure associated to u is concentrated on a 1-rectifiable set, which detects the jump-type discontinuities of u.

Factorization for entropy production of the Eikonal equation and regularity

“…It is expected that such concentration of entropy measures on the H 1 -rectifiable set J, if resolved, will be a crucial step towards the full proof of the Aviles-Giga conjecture. Note that very recently Marconi has resolved the analogous versions of Conjecture 1 for Burgers equation [Mar20b] and the micromagnetics functional [Mar20a] using a powerful Lagrangian representation method.…”

Characterization of minimizers of Aviles-Giga functionals in special domains

“…Although Theorem 1.5 has a perfect analogue for the elements in the asymptotic domain of E ε (see [AKLR02]), the main difficulty seems to be a still not complete understanding of the fine properties of these elements. In this direction we notice that the method used here to establish Proposition 1.7 gives the analogue in this setting of the concentration property (3') (see [Mar20a]).…”

“…The extension to the case where Ω is a W 2,∞ open set does not cause any significant difficulty. In particular the argument proposed in [Mar20a] applies here with trivial modifications and leads to the following partial result: Lemma 2.6. In the setting of Proposition 2.5, let σ ∈ M loc (Ω × R/2πZ) be a locally finite measure satisfying (1.7).…”

“…The existence of a Lagrangian representation for weak solutions with finite entropy production to general conservation laws on the whole (0, T ) × R d has been proved in [Mar19]. The case of bounded domains when Ω is a ball was considered in [Mar20a] for the class of solutions to the eikonal equation arising in [RS03]. The extension to the case where Ω is a W 2,∞ open set does not cause any significant difficulty.…”

“…The key step is the following regularity result. The strategy of the proof is borrowed from [Mar20a], where an analogous statement was proved for the solutions to the eikonal equation arising in the micromagnetics model mentioned in the introduction. We finally observe that in that situation this result is sufficient to establish the concentration property (3'), while it is not the case here.…”