Singular perturbation and the energy of folds

W. Jin, Robert Kohn

Research output: Contribution to journalArticle

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 - δ 2u 2 x - u 2 y) 2 or (1 - |∇u| 2) .

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

Fingerprint

Singular Perturbation
Fold
Boundary conditions
boundary conditions
perturbation
Singularly Perturbed Problem
Energy
Variational Problem
Transverse
Vary
Lower bound
energy

Keywords

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

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mathematics(all)
  • Applied Mathematics
  • Mathematical Physics
  • Statistical and Nonlinear Physics

Cite this

Singular perturbation and the energy of folds. / Jin, W.; Kohn, Robert.

In: Journal of Nonlinear Science, Vol. 10, No. 3, 05.2000, p. 355-390.

Research output: Contribution to journalArticle

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