Upper bounds for singular perturbation problems involving gradient fields

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21 Scopus citations

Abstract

We prove an upper bound for the Aviles-Giga problem, which involves the minimization of the energy Eε (ν) = ε ∫Ω |▽2ν|2 dx + ε-1Ω (1 - |▽ν|2) 2 dx over ν ∈ H2 (Ω), where ε > 0 is a small parameter. Given ν ∈ W1,∞ (Ω) such that ▽ν ∈ BV and |▽ν| = 1 a.e., we construct a family {νε} satisfying: νε → ν in W 1,p (Ω) and Eεε) → 1/3 ∫J▽ν |▽+ν - ▽ -ν|3 dHN-1 as ε goes to 0.

Original languageEnglish
Pages (from-to)1-43
Number of pages43
JournalJournal of the European Mathematical Society
Volume9
Issue number1
DOIs
StatePublished - 1 Jan 2007
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics (all)
  • Applied Mathematics

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