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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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