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Local minimisers and singular perturbations. (English) Zbl 0676.49011

Summary: We construct local minimisers to certain variational problems. The method is quite general and relies on the theory of \(\Gamma\)-convergence. The approach is demonstrated through the model problem \[ \inf \int_{\Omega}\epsilon | \nabla u|^ 2+\epsilon^{-1}(u^ 2- 1)^ 2 dx. \] It is shown that in certain nonconvex domains \(\Omega \subset {\mathbb{R}}^ n\) and for \(\epsilon\) small, there exist nonconstant local minimisers \(u^{\epsilon}\) satisfying \(u^{\epsilon}\approx \pm 1\) except in a thin transition layer. The location of the layer is determined through the requirement that in the limit \(u^{\epsilon}\to u^ 0\), the hypersurface separating the states \(u^ 0=1\) and \(u^ 0=- 1\) locally minimises surface area. Generalisations are discussed with, for example, vector-valued u and “anisotropic” perturbations replacing \(| \nabla u|^ 2\).

MSC:

49J99 Existence theories in calculus of variations and optimal control
49J45 Methods involving semicontinuity and convergence; relaxation
49Q20 Variational problems in a geometric measure-theoretic setting
49K40 Sensitivity, stability, well-posedness
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