Local minimisers and singular perturbations

Abstract

SynopsisWe 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 problemIt 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.

Soliton-Stripe Patterns in Charged Langmuir Monolayers

“…The next result proved by Kohn and Sternberg [10] asserts that as a corollary of Proposition 3.1 near every isolated local minimizer of J there exists a local minimizer of I ǫ . The original result in [10] deals with a domain with a boundary.…”

Diffuse interface methods for multiclass segmentation of high-dimensional data

“…Kohn et al show in [4] that the → 0 limit of (1), in the sense of Γ-convergence, is the total variation semi-norm:…”

From non-linear elasticity to linear elasticity with initial stress via Γ-convergence

“…Before proceeding further on, we point out the following immediate consequences of (H1-H3), (8), and (9): (i) w is a Caratheodory's function; (ii) the function E → w(x, E) is twice continuously differentiable near the origin; (iii) the residual stress…”

“…convergence is not well suited for studying the convergence of local minimizers, even if there are a few papers discussing the issue. The first is due to Kohn and Sternberg [8] where, under appropriate assumptions, the existence and the convergence of local minimizers for a sequence of functionals is determined (under the assumption that the limit functional has a local minimizer). These results apply to quasi-convex integrands with a p-growth from above; it is well known that this last assumption is incompatible with impenetrability of matter, i.e., our assumption (H1).…”