TY - JOUR

T1 - On the Γ-limit of singular perturbation problems with optimal profiles which are not one-dimensional. Part II

T2 - The lower bound

AU - Poliakovsky, Arkady

N1 - Publisher Copyright:
© 2015, Hebrew University of Jerusalem.

PY - 2015/9/1

Y1 - 2015/9/1

N2 - We construct the lower bound, in the spirit of Γ-convergence for some general classes of singular perturbation problems, with or without a prescribed differential constraint, of the form (Formula Presented.) for (Formula Presented.) such that (Formula Presented.), where the function F is nonnegative and A: ℝk×N → ℝm is a prescribed linear operator (for example, A:≡ 0, A · ▿v:= curl v and A · ▿v = divv). Furthermore, we study the cases where we can easily prove that this lower bound coincides with the upper bound obtained in [18]. In particular, we find the formula for the Γ-limit for a general class of anisotropic problems without a differential constraint (i.e., in the case A:≡ 0).

AB - We construct the lower bound, in the spirit of Γ-convergence for some general classes of singular perturbation problems, with or without a prescribed differential constraint, of the form (Formula Presented.) for (Formula Presented.) such that (Formula Presented.), where the function F is nonnegative and A: ℝk×N → ℝm is a prescribed linear operator (for example, A:≡ 0, A · ▿v:= curl v and A · ▿v = divv). Furthermore, we study the cases where we can easily prove that this lower bound coincides with the upper bound obtained in [18]. In particular, we find the formula for the Γ-limit for a general class of anisotropic problems without a differential constraint (i.e., in the case A:≡ 0).

UR - http://www.scopus.com/inward/record.url?scp=84945904403&partnerID=8YFLogxK

U2 - 10.1007/s11856-015-1256-7

DO - 10.1007/s11856-015-1256-7

M3 - Article

AN - SCOPUS:84945904403

VL - 210

SP - 359

EP - 398

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 1

ER -