Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On a singular perturbation problem involving a ``circular-well'' potential


Authors: Nelly André and Itai Shafrir
Journal: Trans. Amer. Math. Soc. 359 (2007), 4729-4756
MSC (2000): Primary 35J20; Secondary 35B25, 35J60, 58E50
DOI: https://doi.org/10.1090/S0002-9947-07-04344-9
Published electronically: May 1, 2007
MathSciNet review: 2320649
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the asymptotic behavior, as a small parameter $ \varepsilon$ goes to 0, of the minimizers for a variational problem which involves a ``circular-well'' potential, i.e., a potential vanishing on a closed smooth curve in $ \mathbb{R}^2$. We thus generalize previous results obtained for the special case of the Ginzburg-Landau potential.


References [Enhancements On Off] (What's this?)

  • 1. N. André and I. Shafrir, Minimization of a Ginzburg-Landau type functional with nonvanishing Dirichlet boundary condition, Calc. Var. Partial Differential Equations 7 (1998), 191-217. MR 1651415 (99k:35161)
  • 2. N. André and I. Shafrir, On a singular perturbation problem involving the distance to a curve, J. d'Anal. Math. 90 (2003), 337-396. MR 2001075 (2004h:35052)
  • 3. N. André and I. Shafrir, On the minimizers of a Ginzburg-Landau type energy when the boundary condition has zeros, Advances in Diff. Equations 9 (2004), 891-960. MR 2100398 (2005g:35050)
  • 4. F. Bethuel, H. Brezis and F. Hélein, Asymptotics for the minimization of a Ginzburg-Landau functional, Calc. Var. Partial Differential Equations 1 (1993),123-148. MR 1261720 (94m:35083)
  • 5. F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices, Birkhäuser, 1994. MR 1269538 (95c:58044)
  • 6. L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998. MR 1625845 (99e:35001)
  • 7. I. Fonseca and L. Tartar, The gradient theory of phase transitions for systems with two potential wells, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), 89-102. MR 985992 (90b:49021)
  • 8. D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin and New York, 1983. MR 737190 (86c:35035)
  • 9. L. Lassoued and P. Mironescu, Ginzburg-Landau type energy with discontinuous constraint, J. Anal. Math. 77 (1999), 1-26. MR 1753480 (2001e:58014)
  • 10. L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal. 98 (1987), 123-142. MR 866718 (88f:76038)
  • 11. P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal. 101 (1988), 209-260. MR 930124 (89h:49007)
  • 12. M. Struwe, On the asymptotic behavior of minimizers of the Ginzburg-Landau model in 2 dimensions, Differential Integral Equations 7 (1994), 1613-1624; erratum, loc. cit. 8 (1995), 124. MR 1269674 (95g:35057a)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35J20, 35B25, 35J60, 58E50

Retrieve articles in all journals with MSC (2000): 35J20, 35B25, 35J60, 58E50


Additional Information

Nelly André
Affiliation: Département de Mathématiques, Université de Tours, 37200 Tours, France

Itai Shafrir
Affiliation: Department of Mathematics, Technion – Israel Institute of Technology, 32000 Haifa, Israel

DOI: https://doi.org/10.1090/S0002-9947-07-04344-9
Keywords: Singular perturbation, circular-well potential, Ginzburg-Landau energy
Received by editor(s): April 5, 2005
Published electronically: May 1, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society