Variational problems with singular perturbation

Abstract

In this paper, we construct the local minimum of a certain variational problem which we take in the form inf ∫ Ω { ϵ 2 k g 2 | ∇ w | 2 + 1 4 ϵ f 2 g 4 ( 1 − w 2 ) 2 } d x , where ϵ is a small positive parameter and Ω ⊂ R n is a convex bounded domain with smooth boundary. Here f , g , k ∈ C 3 ( Ω ) are strictly positive functions in the closure of the domain Ω ̄ . If we take the inf over all functions H 1 ( Ω ) , we obtain the (unique) positive solution of the partial differential equation with Neumann boundary conditions (respectively Dirichlet boundary conditions). We wish to restrict the inf to the local (not global) minimum so that we consider solutions of this Neumann problem which take both signs in Ω and which vanish on ( n − 1 ) dimensional hypersurfaces Γ ϵ ⊂ Ω . By using a Γ -convergence method, we find the structure of the limit solutions as ϵ → 0 in terms of the weighted geodesics of the domain Ω .