Upper bounds for singular perturbation problems involving gradient fields

Arkady Poliakovsky

doi:10.4171/JEMS/70

Upper bounds for singular perturbation problems involving gradient fields

Arkady Poliakovsky

Research output: Contribution to journal › Review article › peer-review

31 Scopus citations

Abstract

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

Original language	English
Pages (from-to)	1-43
Number of pages	43
Journal	Journal of the European Mathematical Society
Volume	9
Issue number	1
DOIs	https://doi.org/10.4171/JEMS/70
State	Published - 1 Jan 2007
Externally published	Yes

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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title = "Upper bounds for singular perturbation problems involving gradient fields",

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-1 ∫Ω (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 Eε (νε) → 1/3 ∫J▽ν |▽+ν - ▽ -ν|3 dHN-1 as ε goes to 0.",

author = "Arkady Poliakovsky",

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doi = "10.4171/JEMS/70",

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AU - Poliakovsky, Arkady

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N2 - 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-1 ∫Ω (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 Eε (νε) → 1/3 ∫J▽ν |▽+ν - ▽ -ν|3 dHN-1 as ε goes to 0.

AB - 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-1 ∫Ω (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 Eε (νε) → 1/3 ∫J▽ν |▽+ν - ▽ -ν|3 dHN-1 as ε goes to 0.

UR - https://www.scopus.com/pages/publications/33845484043

U2 - 10.4171/JEMS/70

DO - 10.4171/JEMS/70

M3 - Review article

AN - SCOPUS:33845484043

SN - 1435-9855

VL - 9

SP - 1

EP - 43

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

IS - 1

ER -

Upper bounds for singular perturbation problems involving gradient fields

Abstract

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