Geodesic
Upper bounds for singular perturbation problems involving gradient fields
Journal of the European Mathematical Society, Tome 9 (2007) no. 1, pp. 1-43.

Voir la notice de l'article provenant de la source EMS Press

We prove an upper bound for the Aviles–Giga problem, which involves the minimization of the energy Eε​(v)=ε∫Ω​​∇2v​2dx+ε1​∫Ω​(1−∣∇v∣2)2dx over v∈H2(Ω), where ε>0 is a small parameter. Given v∈W1,∞(Ω) such that ∇v∈BV and ∣∇v∣=1 a.e., we construct a family {vε​} satisfying: vε​→v in W1,p(Ω) and Eε​(vε​)→31​∫J∇v​​∣∇+v−∇−v∣3dHN−1, as ε goes to 0.
DOI : 10.4171/jems/70
Classification : 49-XX, 00-XX
Keywords: 
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     title = {Upper bounds for singular perturbation problems involving gradient fields},
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Arkady Poliakovsky. Upper bounds for singular perturbation problems involving gradient fields. Journal of the European Mathematical Society, Tome 9 (2007) no. 1, pp. 1-43. doi : 10.4171/jems/70. http://geodesic.mathdoc.fr/articles/10.4171/jems/70/

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