English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

where G(u) = u2(1 − u)2 is a double well potential, ε > 0, and Aε(x) = 1+A(x)/ε with 0 ≤ A ∈ C1(Rn), 1−periodic in x1, · · · , xn, Ω is the support of A|[0,1]n and has a smooth boundary, and Ω ⊂ (0, 1)n. A main result of [9] is that there is an ε0 > 0 such that for any finite set T ⊂ Zn and ε ∈ (0, ε0], (1.1) has a solution, Uε,T with 0 < Uε,T < 1, Uε,T is near 1 on A T ≡ T +Ω and near 0 on BT ≡ (Zn\T )+Ω. Moreover as ε → 0, Uε,T → 1 uniformly on AT and Uε,T → 0 uniformly on BT . When T is finite, Uε,T is characterized as the minimizer of a constrained variational problem associated with (1.1). Although Uε,T may not be unique, the set of such minimizers, Mε(T ), is ordered. The setting of [9] was further treated in [10] where it was shown that for each finite T , there is an ε1(T ) > 0 such that for ε ∈ (0, ε1(T )), (1.1) has a solution, Vε,T of mountain pass type with 0 < Vε,T < Uε,T . The main goal of this note is to study the setting of when T is finite and consists of two widely separated subsets, that is, T = T1 ∪ (l + T2) ≡ Tl for T1, T2 ⊂ Zn, l ∈ Zn and large |l| > 0. In particular we are interested in the asymptotic behavior as l → ∞ of the minimizers, Uε,Tl , and the mountain pass solutions, as well as the corresponding critical values. To describe our

Asymptotic behavior of minima and mountain pass solutions for a class of Allen-Cahn models