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Local minimisers and singular perturbations*

Published online by Cambridge University Press:  14 November 2011

Robert V. Kohn  and
Peter Sternberg
Robert V. Kohn
Affiliation:
Courant Institute, 251 Mercer Street, New York, NY 10012, U.S.A.
Peter Sternberg
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, U.S.A.
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Synopsis

We construct local minimisers to certain variational problems. The method is quite general and relies on the theory of Γ-convergence. The approach is demonstrated through the model problem

It is shown that in certain nonconvex domains Ω ⊂ ℝn and for ε small, there exist nonconstant local minimisers uε satisfying uε ≈ ± 1 except in a thin transition layer. The location of the layer is determined through the requirement that in the limit uε →u0, the hypersurface separating the states u0 = 1 and u0 = −1 locally minimises surface area. Generalisations are discussed with, for example, vector-valued u and “anisotropic” perturbations replacing |∇u|2.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 111 , Issue 1-2 , 1989 , pp. 69 - 84
Copyright
Copyright © Royal Society of Edinburgh 1989

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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 modeling phase transitions. SIAM J. Math. Anal. 18 (1987), 1453–1462.CrossRefGoogle Scholar
3Alikakos, N. and Simpson, H.. A variational approach for a class of singular perturbation problems and applications. Proc. Roy. Soc. Edinburgh Sect. A 107 (1987), 27–42.CrossRefGoogle Scholar
4Almgren, F. J.. Existence and Regularity Almost Everywhere of Solutions to Elliptic Variational Problems with Constraints, Mem. Amer. Math. Soc. 165 (Providence R.I.: American Mathematical Society).Google Scholar
5Angenent, S. B., Mallet-Paret, J. and Peletier, L. A.. Stable transition layers in a semilinear boundary value problem. J. Differential Equations 67 (1987), 212–242.CrossRefGoogle Scholar
6Berger, M. S. and Fraenkel, L. E.. On the asymptotic solution of a nonlinear Dirichlet problem. J. Math. Mech. 19 (1969–1970), 553–585.Google Scholar
7Caginalp, G.. Solidification problems as systems of nonlinear differential equations. In Nonlinear Systems of Partial Differential Equations in Applied Math., AMS Lectures in Appl. Math., Vol. 23, part 2, pp. 347–370 (Providence, Rhode Island: American Mathematical Society).Google Scholar
8Caginalp, G. and Fife, P. C.. An analysis of a phase field model of a free boundary. Arch. Rational Mech. Anal. 92 (1986), 205–242.CrossRefGoogle Scholar
9Carr, J., Gurtin, M. E., and Slemrod, M.. Structured phase transitions on a finite interval. Arch. Rational Mech. Anal. 86 (1984), 317–351.CrossRefGoogle Scholar
10Casten, R. G. and Holland, C. J.. Instability results for reaction diffusion equations with Neumann boundary conditions. J. Differential Equations 27 (1978), 266–273.CrossRefGoogle Scholar
11Chafee, N.. Asymptotic behavior for solutions of a one-dimensional parabolic equation with homogeneous Neumann boundary conditions. J. Differential Equations 18 (1975), 111–134.CrossRefGoogle Scholar
12Conway, E., Hoff, D. and Smoller, J.. Large time behavior of solutions of systems of nonlinear reaction-diffusion equations. SIAM J. Appl. Math. 35 (1978), 1–16.CrossRefGoogle Scholar
13Dacorogna, B.. Weak Continuity and Weak Lower Semicontinuity of Nonlinear Functionals. Lecture Notes in Mathematics 922 (Berlin: Springer, 1982).Google Scholar
14Maso, G. Dal and Modica, L.. Sulla convergenza dei minimi locali. Boll. Un. Mat. Ital (6) 1A (1982), 55–61.Google Scholar
15Giorgi, E. De. Convergence problems for functionals and operators. In Proc. Int. Meeting on Recent Methods in Nonlinear Analysis, eds. Giorgi, E. D. et al. , pp. 223–244 (Bologna: Pitagora, 1979).Google Scholar
16Fife, P. C. and Greenlee, W. M.. Interior transition layers for elliptic boundary value problems. Russian Math. Surveys 29 (1974), 103–131.CrossRefGoogle Scholar
17Giusti, E.. Minimal surfaces and functions of bounded variation (Basel: Birkhauser, 1984).CrossRefGoogle Scholar
18Gurtin, M. E.. On a theory of phase transitions with interfacial energy. Arch. Rational Mech. Anal. 87 (1984), 187–212.CrossRefGoogle Scholar
19Gurtin, M. E.. Some results and conjectures in the gradient theory of phase transitions. In Metastability and Incompletely Posed Problems, eds. Antman, S. et al. , pp. 135–146 (Berlin: Springer, 1987).Google Scholar
20Gurtin, M. E. and Matano, H.. On the structure of equilibrium phase transitions within the gradient theory of fluids. Quart. Appl. Math. 46 (1988), 301–317.CrossRefGoogle Scholar
21Hale, J. and Vegas, J.. A nonlinear parabolic equation with varying domain. Arch. Rational Mech. Anal. 86 (1984), 99–123.CrossRefGoogle Scholar
22Mahoney, J. J. and Norbury, J.. Asymptotic location of nodal lines using geodesic theory. J. Austral. Math. Soc. Ser. B 27 (1986), 259–280.CrossRefGoogle Scholar
23Massari, U. and Miranda, M.. Minimal surfaces of codimension one (Amsterdam: North Holland, 1984).Google Scholar
24Matano, H.. Asymptotic behavior and stability of solutions of semilinear diffusion equations. Publ. Res. Inst. Math. Sci. 15 (1979), 401–424.CrossRefGoogle Scholar
25Matano, H. and Mimura, M.. Pattern formation in competition-diffusion systems in nonconvex domains. Publ. Res. Inst. Math. Sci. 19 (1983), 1049–1079.CrossRefGoogle Scholar
26Modica, L.. Γ-convergence to minimal surfaces problem and global solutions of Δu = 2(u3 - u). In Proc. Int. Meeting on Recent Methods in Nonlinear Analysis, eds. Giorgi, E. de et al. , pp. 223–244 (Bologna: Pitagora, 1979).Google Scholar
27Modica, L.. Gradient theory of phase transitions and minimal interface criteria. Arch. Rational Mech. Anal. 98 (1987), 123–142.CrossRefGoogle Scholar
28Modica, L.. Gradient theory of phase transitions with boundary contact energy. Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1987), 453–486.Google Scholar
29Modica, L. and Mortola, S.. II limite nella Γ-convergenza di una famiglia di funzionali ellittichi. Boll. Un. Math. Ital. A (3) 14 (1977), 526–529.Google Scholar
30Modica, L. and Mortola, S.. Un esempio di Γ-convergenza. Boll. Un. Mat. Ital. (5) 14B (1977), 285–299.Google Scholar
31Modica, L. and Luckhaus, S.. The Gibbs-Thompson relation within the gradient theory of phase transitions (to appear).Google Scholar
32Murray, J. D.. On pattern formation mechanism for lepidopheran wing patterns and mammalian coat markings. Philos. Trans. Roy. Soc. London Ser. B 295 (1981), 473–496.Google ScholarPubMed
33Murray, J. D.. Prepattern formation mechanism for animal coat markings. J. Theoret. Biol. 88 (1981), 161–199.CrossRefGoogle Scholar
34Novick-Cohen, A. and Segel, L. A.. Nonlinear aspects of the Cahn-Hilliard equation. Phys. D 10 (1984), 278–298.Google Scholar
35Owen, N. C.. Nonconvex variational problems with general singular perturbations. Trans. Amer. Math. Soc. (to appear).Google Scholar
36Sternberg, P.. The effect of a singular perturbation on nonconvex variational problems. (Ph.D. Thesis, New York University, 1986).Google Scholar
37Sternberg, P.. The effect of a singular perturbation on nonconvex variational problems. Arch Rational Mech. Anal. 101 (1988), 209–260.CrossRefGoogle Scholar
38Vegas, J. M.. Bifurcations caused by perturbing the domain in an elliptic equation. J. Differential Equations 48 (1983), 189–226.CrossRefGoogle Scholar
39Vegas, J. M.. A Neumann elliptic problem with variable domain. In Contributions to Nonlinear Partial Differential Equations, eds. Bardos, C. et al. , pp. 264–273 (London: Pitman, 1983).Google Scholar
40Sternberg, P.. Vector-valued local minimizers of nonconvex variational problems. In Proc. of Workshop on Nonlinear P.D.E.'s, Provo, 1987, eds. Bates, P. and Fife, P. (Berlin: Springer, to appear).Google Scholar
41Owen, N. C. and Sternberg, P.. Nonconvex variational problems with anisotropic perturbations (preprint).Google Scholar
42Fonseca, I. and Tartar, L.. The gradient theory of phase transitions for systems with two potential wells. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), 89–102.CrossRefGoogle Scholar
43Ball, J. and James, R. D.. Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987), 13–52.CrossRefGoogle Scholar