Upper bounds for singular perturbation problems involving gradient fields

Sharp upper bounds for a variational problem with singular perturbation

Let Ω be a C bounded open set of R and consider the functionals Fε(u) := ∫ Ω { (1 − |∇u(x)|2)2 ε + ε|Du(x)| } dx . We prove that if u ∈ W (Ω), |∇u| = 1 a.e., and ∇u ∈ BV , then Γ− lim ε↓0 Fε(u) = 1 3 ∫ J∇u |[∇u]|dH 1 . The new result is the Γ− lim sup inequality.

A General Technique to Prove Upper Bounds for Singular Perturbation Problems

* Firstly, we wish to find a lower bound, i.e. the functional I(φ) such that for every family {φε}ε>0, satisfying φε → φ as ε → 0 , we have limε→0+ Iε(φε) ≥ I(φ). ** Secondly, we wish to find an upper bound, i.e. the functional I(φ) such that there exists the family {ψε}ε>0, satisfying ψε → φ as ε → 0 , and we have limε→0+ Iε(ψε) ≤ I(φ). *** If we obtain I(φ) = I(φ) := I(φ), then I(φ) will be t...

Sharp Upper Bounds for a Singular Perturbation Problem Related to Micromagnetics

We construct an upper bound for the following family of functionals {Eε}ε>0, which arises in the study of micromagnetics: Eε(u) = ∫ ε|∇u|2 + 1 ε ∫ R2 |Hu |2. Here is a bounded domain in R2, u ∈ H1( , S1) (corresponding to the magnetization) and Hu , the demagnetizing field created by u, is given by { div (ũ + Hu) = 0 in R2 , curl Hu = 0 in R2 , where ũ is the extension of u by 0 in R2 \ . Our u...

Perturbation bounds for eigenvalues of diagonalizable matrices and singular values

On some anisotropic singular perturbation problems

We investigate the asymptotic behavior of some anisotropic diffusion problems and give some estimates on the rate of convergence of the solution toward its limit. We also relate this type of elliptic problems to problems set in cylinder becoming unbounded in some directions and show how some information on one type leads to information for the other type and conversely. 1. A model problem The g...