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Local minimisers and singular perturbations*

  • Robert V. Kohn (a1) and Peter Sternberg (a2)
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.

Copyright
COPYRIGHT: © Royal Society of Edinburgh 1989
References
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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
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  • EISSN: 1473-7124
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