Singular perturbation and the energy of folds

W. Jin, R. V. Kohn

Research output: Contribution to journalArticlepeer-review

Abstract

We address the singularly perturbed variational problem ∫ ∈-1(1-|∇u|2)2+ ∈|∇∇u|2 in two space dimensions. We introduce a new scheme for proving lower bounds and show the bounds are asymptotically sharp for certain domains and boundary conditions. Our results support the conjecture, due to Aviles and Giga, that folds are one-dimensional, i.e., ∇u varies mainly in the direction transverse to the fold. We also consider related problems obtained when (1 - |∇u|2)2 is replaced by (1 - δ2u2x - u2y)2 or (1 - |∇u|2).

Original languageEnglish (US)
Pages (from-to)355-390
Number of pages36
JournalJournal of Nonlinear Science
Volume10
Issue number3
DOIs
StatePublished - 2000

Keywords

  • Fold energy
  • Gamma - convergence
  • Lower bounds
  • Singular perturbation
  • Transition layers
  • Viscosity solution

ASJC Scopus subject areas

  • Modeling and Simulation
  • Engineering(all)
  • Applied Mathematics

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