Variational problems with singular perturbation

Yeh, Li-Chin and Norbury, John (2005) Variational problems with singular perturbation. Nonlinear Analysis . (In Press)

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Official URL: http://dx.doi.org/10.1016/j.na.2005.11.009

Abstract

In this paper, we construct the local minimum of a certain variational problem which we take in the form

,

where is a small positive parameter and is a convex bounded domain with smooth boundary. Here are strictly positive functions in the closure of the domain . If we take the inf over all functions , 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 and which vanish on dimensional hypersurfaces . By using a -convergence method, we find the structure of the limit solutions as in terms of the weighted geodesics of the domain .

Item Type: Article O - Z > Partial differential equationsA - C > Calculus of variations and optimal control Oxford Centre for Industrial and Applied Mathematics 308 Gareth Wyn Jones 03 Nov 2006 29 May 2015 18:19

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