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A variational approach for a class of singular perturbation problems and applications

Published online by Cambridge University Press:  14 November 2011

Nicholas D. Alikakos  and
Henry C. Simpson
Nicholas D. Alikakos
Affiliation:
Mathematics Department, University of Tennessee, Knoxville, TN 37996, U.S.A.; Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University, Providence, RI 02912, U.S.A.
Henry C. Simpson
Affiliation:
Mathematics Department, University of Tennessee, Knoxville, TN 37996, U.S.A.
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Synopsis

We study the limit as ε → 0 of global minimisers of functionals of the type

where Ω is an annul us or a ball in ℝn.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 107 , Issue 1-2 , 1987 , pp. 27 - 42
Copyright
Copyright © Royal Society of Edinburgh 1987

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References

1Alikakos, N. D. and Bates, P.. On the singular limit in a phase field model of phase transitions. Ann. Inst. H. Poincaré: Anal. Non lineaire (to appear).Google Scholar
2Alikakos, N. D. and Shaing, K. C.. On the singular limit for a class of problems modelling phase transitions. SIAM J. Math. Anal, (to appear).Google Scholar
3Alikakos, N. D. and Simpson, H.. Uniqueness of solutions for a class of problems modelling phase transitions (unpublished manuscript).Google Scholar
4Angenent, S. B.. Uniqueness of the solution of a semilinear boundary value problem (preprint).Google Scholar
5Angenent, S. B., Mallet-Paret, J. and Peletier, L. A.. Stable transition layers in a semilinear boundary value problem (preprint).Google Scholar
6Caginalp, G. and Fife, P. C.. Elliptic problems involving phase boundaries satisfying a curvature condition (preprint).Google Scholar
7Carr, J., Gurtin, M. E. and Slemrod, M.. Structured phase transitions on a finite interval. Arch. Rational Mech. Anal. 86 (1984), 317–351.CrossRefGoogle Scholar
8Clement, Ph. and Peletier, L. A.. On a nonlinear eigenvalue problem occurring in population genetics. Proc. Roy. Soc. Edinburgh Sect. A 100 (1985), 85–101.CrossRefGoogle Scholar
9Fife, P. C.. Transition layers in singular perturbation problems. J. Differential Equations 15 (1974), 77–105.CrossRefGoogle Scholar
10Fife, P. C.. Boundary and interior layer phenomena for pairs of second-order differential equations. J. Math. Anal. Appl. 54 (1976), 497–521.CrossRefGoogle Scholar
11Fife, P. C. and Greenlee, W. M.. Interior transition layers for elliptic boundary value problems with small parameter. Russian Math. Surveys 29 (1974), 103–131.CrossRefGoogle Scholar
12Friedman, A. and Phillips, D.. The free boundary of a semilinear elliptic equation. Trans. Amer. Math. Soc. 282 (1984), 153–182.CrossRefGoogle Scholar
13Gurtin, M.. Some results and conjectures in the gradient theory of phase transitions (IMA preprint 156).Google Scholar
14Henry, D.. Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics 840 (Berlin: Springer, 1981).Google Scholar
15Kawohl, B.. (personal communication.)Google Scholar
16Nishiura, V. and Fuji, H.. Stability of theorem for singularly perturbed solutions to systems of reaction-diffusion equations. Proc. Japan Acad. Ser. A Math. Sci. 61 (1985), 329–332.CrossRefGoogle Scholar
17Nishiura, V. and Fuji, H.. SLEP method to the stability of singularly perturbed solutions with multiple interval transition layers in Reaction-diffusion systems. Proceedings of NATO Advanced Research Workshop, Dynamics of Infinite-dimensional systems, eds. Hale, J. K. & Chow, S. N. (Berlin: Springer, to appear).Google Scholar
18Owen, N.. Existence and stability of necking deformations for nonlinearly elastic rods (LCDS Report 86–12, Brown University).CrossRefGoogle Scholar
19Smoller, J.. Shock waves and reaction-diffusion equations. (Berlin: Springer, 1982).Google Scholar
20Natanson, V. I.. Theory of functions of a real variable, vol. I, English translation (Ungar).Google Scholar