Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Local minimisers and singular perturbations*

Robert V. Kohna1 and Peter Sternberga2 p1

a1 Courant Institute, 251 Mercer Street, New York, NY 10012, U.S.A.

a2 Department of Mathematics, Stanford University, Stanford, CA 94305, U.S.A.

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

S0308210500025026_eqnU1

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 uεu0, 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.

(Received July 27 1987)

(Revised April 21 1988)

Correspondence

p1 Present address: Department of Mathematics, Indiana University, Bloomington, IN 47405, U.S.A.

Footnotes

* This work was supported in part by NSF grant DMS-8312229, ONR grant N00014-83-0536, and the Sloan Foundation.