A Nonlocal Problem Arising in the Study of Magneto-Elastic Interactions

M. Chipot; I. Shafrir; G. Vergara Caffarelli

Bollettino dell'Unione Matematica Italiana (2008)

  • Volume: 1, Issue: 1, page 197-221
  • ISSN: 0392-4041

Abstract

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The energy of magneto-elastic materials is described by a nonconvex functional. Three terms of the total free energy are taken into account: the exchange energy, the elastic energy and the magneto-elastic energy usually adopted for cubic crystals. We focus our attention to a one dimensional penalty problem and study the gradient flow of the associated type Ginzburg-Landau functional. We prove the existence and uniqueness of a classical solution which tends asymptotically for subsequences to a stationary point of the energy functional.

How to cite

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Chipot, M., Shafrir, I., and Vergara Caffarelli, G.. "A Nonlocal Problem Arising in the Study of Magneto-Elastic Interactions." Bollettino dell'Unione Matematica Italiana 1.1 (2008): 197-221. <http://eudml.org/doc/290458>.

@article{Chipot2008,
abstract = {The energy of magneto-elastic materials is described by a nonconvex functional. Three terms of the total free energy are taken into account: the exchange energy, the elastic energy and the magneto-elastic energy usually adopted for cubic crystals. We focus our attention to a one dimensional penalty problem and study the gradient flow of the associated type Ginzburg-Landau functional. We prove the existence and uniqueness of a classical solution which tends asymptotically for subsequences to a stationary point of the energy functional.},
author = {Chipot, M., Shafrir, I., Vergara Caffarelli, G.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {197-221},
publisher = {Unione Matematica Italiana},
title = {A Nonlocal Problem Arising in the Study of Magneto-Elastic Interactions},
url = {http://eudml.org/doc/290458},
volume = {1},
year = {2008},
}

TY - JOUR
AU - Chipot, M.
AU - Shafrir, I.
AU - Vergara Caffarelli, G.
TI - A Nonlocal Problem Arising in the Study of Magneto-Elastic Interactions
JO - Bollettino dell'Unione Matematica Italiana
DA - 2008/2//
PB - Unione Matematica Italiana
VL - 1
IS - 1
SP - 197
EP - 221
AB - The energy of magneto-elastic materials is described by a nonconvex functional. Three terms of the total free energy are taken into account: the exchange energy, the elastic energy and the magneto-elastic energy usually adopted for cubic crystals. We focus our attention to a one dimensional penalty problem and study the gradient flow of the associated type Ginzburg-Landau functional. We prove the existence and uniqueness of a classical solution which tends asymptotically for subsequences to a stationary point of the energy functional.
LA - eng
UR - http://eudml.org/doc/290458
ER -

References

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  1. BERTSCH, M. - PODIO-GUIDUGLI, P. - VALENTE, V., On the dynamics of deformable ferromagnets. I. Global weak solutions for soft ferromagnets at rest, Ann. Mat. Pura Appl., 179 (2001), 331-360. Zbl1097.74017MR1848770DOI10.1007/BF02505962
  2. BETHUEL, F. - BREZIS, H. - COLEMAN, B. D. - HÉLEIN, F., Bifurcation analysis of minimizing harmonic maps describing the equilibrium of nematic phases between cylinders, Arch. Ration. Mech. Anal., 118 (1992), 149-168. Zbl0825.76062MR1158933DOI10.1007/BF00375093
  3. BROWN, W. F., Micromagnetics, John Wiley and Sons (Interscience), 1963. 
  4. BROWN, W. F., Magnetoelastic Interactions, Springer Tracts in Natural Philosophy, 9, Springer Verlag, 1966. 
  5. DESIMONE, A. - DOLZMANN, G., Existence of minimizers for a variational problem in two-dimensional nonlinear magnetoelasticity, Arch. Rational Mech. Anal., 144 (1998), 107-120. Zbl0923.73079MR1657391DOI10.1007/s002050050114
  6. DESIMONE, A. - JAMES, R. D., A constrained theory of magnetoelasticity, J. Mech. Phys. Solids, 50 (2002), 283-320. Zbl1008.74030MR1892979DOI10.1016/S0022-5096(01)00050-3
  7. HE, S., Modélisation et simulation numérique de matériaux magnétostrictifs, PhD thesis, Université Pierre et Marie Currie, 1999. 
  8. KINDERLEHRER, D., Magnetoelastic interactions. Variational methods for discontinuous structures, Prog. Nonlinear Differential Equations Appl., BirkhauserBasel, 25, (1996), 177-189. MR1414500
  9. VALENTE, V. - VERGARA CAFFARELLI, G., On the dynamics of magneto-elastic interations: existence of weak solutions and limit behaviors, Asymptotic Analysis, 51 (2007), 319-333. Zbl1125.35100MR2321728

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