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minimizers for the hartree-fock-bogoliubov theory of neutron stars ...

minimizers for the hartree-fock-bogoliubov theory of neutron stars ...

296 LENZMANN and LEWIN

296 LENZMANN and LEWIN Proof We consider a smooth partition of unity {χ k } k∈Z 3 with ∑ k∈Z 3 χ 2 k ≡ 1 such that χ k ≡ 1 on C k and supp χ k ⊂ C ′ k , with the half-open cubes C k = [k, k + 1) 3 and C ′ k = [k − 1,k+ 2) 3 where k ∈ Z 3 . Furthermore, we assume that sup k∈Z 3 |∇χ k (x)| C. Next, we estimate as follows: ∫R 3 |f n (x)| 8/3 dx = ∑ k∈Z 3 ∫ R 3 χ k (x) 2 |f n (x)| 8/3 dx = ∑ k∈Z 3 ∫R 3 |1 C ′ k (x)f n (x)| 2/3 |χ k (x)f n (x)| 2 dx ∑ k∈Z 3 ‖1 C ′ k |f n | 2 ‖ 1/3 L 1 (R 3 ) ‖χ kf n ‖ 2 L 3 (R 3 ) C sup k∈Z 3 ‖1 C ′ k |f n | 2 ‖ 1/3 L 1 (R 3 ) ∑ k∈Z 3 〈f n ,χ k Kχ k f n 〉. (7.4) We have used the Sobolev-type inequality ‖f ‖ 2 L 3 (R 3 ) C〈f, Kf 〉. By Lemma A.1, the nonlocal operator K satisfies ∑ χ k Kχ k K + 1 ∫ ∞ 1 ( ∑ ) |∇χ π k∈Z 3 0 s + K 2 k | 2 1 √ sds K + C, (7.5) s + K 2 k∈Z 3 since the integral expression on the right-hand side in the first inequality is a bounded self-adjoint operator due to the fact that ∑ k∈Z 3 |∇χ k(x)| 2 ∈ L ∞ (R 3 ) by our choice of the partition {χ k } k∈Z 3. Therefore, we conclude that ∑ k∈Z 3 〈f n ,χ k Kχ k f n 〉 C‖f n ‖ 2 H 1/2 (R 3 ) . (7.6) Since, by assumption, we have ‖f n ‖ H 1/2 (R 3 ) C independent of n, estimate (7.4) leads to ∫ |f n (x)| 8/3 dx C sup‖1 C ′ k |f n | 2 ‖ 1/3 L 1 (R 3 ) → 0, (7.7) R 3 k∈Z 3 using that {|f n | 2 } n∈N vanishes. This shows that f n → 0 in L 8/3 (R 3 ), whence it converges to 0 strongly in L p (R 3 ) for all 2

MINIMIZERS FOR THE HFB-THEORY 297 Proof of Lemma 7.1 By the coercivity of E stated in Lemma 4.2, we know that our minimizing sequence (γ n ,α n ) is bounded in X. Hence we deduce from Lemma 2.1 that √ ρ γn is bounded in H 1/2 (R 3 ). Applying Lemma 7.2 to the sequence f n = √ ρ γn , we obtain that ρ γn → 0 strongly in L p (R 3 ) for 1

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