A viscosity property of minimizing micromagnetic configurations

Ambrosio, Luigi; Lecumberry, Myriam; Rivière, Tristan

doi:10.1002/cpa.10070

Cited by 16 publications

(17 citation statements)

References 12 publications

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“…Our twinning example had the wall energy built into the functional, so the analogue of Fact 0 was automatic there. In the present setting we must instead argue as for (3). Since 2xy ≤ x 2 + y 2 we have…”

Section: Branching Of Magnetic Domainsmentioning

confidence: 98%

“…In writing (12) we implicitly assumed that ∇u ε remains compact, so |∇u 0 | = 1 in the limit; this is true, but the proof is far from trivial [2], [19]. The introduction of entropies and the analogy with conservation laws has led to a lot of progress on this and related problems, including [3], [6], [18], [36], [37], [38], [45], [56], [57]. But the subject is far from finished.…”

Section: The Aviles-giga Problemmentioning

confidence: 99%

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Energy-driven pattern formation

Kohn¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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Abstract. Many physical systems can be modelled by nonconvex variational problems regularized by higher-order terms. Examples include martensitic phase transformation, micromagnetics, and the Ginzburg-Landau model of nucleation. We are interested in the singular limit, when the coefficient of the higher-order term tends to zero. Our attention is on the internal structure of walls, and the character of microstructure when it forms. We also study the pathways of thermally-activated transitions, modeled via the minimization of action rather than energy. Our viewpoint is variational, focusing on matching upper and lower bounds. Mathematics Subject Classification (2000). Primary 49-02, 82-02; Secondary 74N15, 82B24, 82D40.

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Section: Branching Of Magnetic Domainsmentioning

confidence: 98%

Section: The Aviles-giga Problemmentioning

confidence: 99%

Energy-driven pattern formation

Kohn¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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show abstract

“…Condition (2.8) follows from the fact that u+ < u~ H1-a.e. on Ju (only decreasing jumps), (2)(3)(4)(5)(6)(7)(8)(9) which itself is a consequence of h being a viscosity solution.…”

Section: Secondmentioning

confidence: 99%

“…Our proof has been inspired by a recent result of Ambrosio, Lecumberry, and Riviere [2]. There the authors proved that, if h £ Wla'^(R2) solves the eikonal equation, then a certain one-parameter family of entropy conditions 011 V/i is sufficient to ensure that h is a viscosity solution.…”

Section: Introductionmentioning

confidence: 96%