Concentration on minimal submanifolds for a singularly by Mahmoudi F., Malchiodi A.

By Mahmoudi F., Malchiodi A.

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Extra resources for Concentration on minimal submanifolds for a singularly perturbed neumann problem

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L=Cε−k +1 βl ψl . Since J = −∆N K + O(1), for any integer m one finds ≤ (Jm ϕ, ϕ)L2 (K) m N m−1 ≤ ((−∆N ϕ, ϕ)L2 (K;N K) + (ϕ, ϕ)L2 (K;N K) . K ) ϕ, ϕ)L2 (K;N K) + Cm ((−∆K ) Since ϕ = −k 1 2 Cε l=0 β˜l ϕl , from (106) we deduce that m ≤ (J ϕ, ˜ ϕ) ˜ L2 (K;N K) ε−2m ϕ C 2 ≤ (109) 2m k C 2 2 L2 (K;N K) 2m k + O(ε−2(m−1) ) ϕ  1 −k  2 L2 (K;N K) 2 Cε ε −2m + O(ε −2(m−1) β˜l2  . )  l=0 On the other hand, since in the basis (ψl )l , the function ϕ˜ has non zero components only when l ≥ Cε−k , by the Weyl’s asymptotic formula we have also that  ∞ m 2   l=Cε−k +1 µl βl ; (110) (Jm ϕ, ˜ ϕ) ˜ L2 (K;N K) ≥  ∞  CC 2m k ε−2m l=Cε−k +1 βl2 .

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