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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References:

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