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

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

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

MINIMIZERS FOR THE HARTREE-FOCK-BOGOLIUBOV THEORY OF NEUTRON STARS AND WHITE DWARFS ENNO LENZMANN and MATHIEU LEWIN Abstract We prove the existence of minimizers for Hartree-Fock-Bogoliubov (HFB) energy functionals with attractive two-body interactions given by Newtonian gravity. This class of HFB functionals serves as a model problem for self-gravitating relativistic Fermi systems, which are found in neutron stars and white dwarfs. Furthermore, we derive some fundamental properties of HFB minimizers such as a decay estimate for the minimizing density. A decisive feature of the HFB model in gravitational physics is its failure of weak lower semicontinuity. This fact essentially complicates the analysis compared to the well-studied Hartree-Fock theories in atomic physics. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 2. Basic properties of HFBenergy . . . . . . . . . . . . . . . . . . . . . . 263 3. Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 4. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 5. Proof of Theorem2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 6. Proof of Theorem3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 7. Proof of Theorem1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 8. Proof of Theorem4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 A. Localization ofkinetic energy . . . . . . . . . . . . . . . . . . . . . . 305 B. Proof of Proposition 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 307 C. Proof of Proposition 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 309 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 DUKE MATHEMATICAL JOURNAL Vol. 152, No. 2, c○ 2010 DOI 10.1215/00127094-2010-013 Received 27 September 2008. Revision received 3 August 2009. 2000 Mathematics Subject Classification. Primary 35Q55; Secondary 49J40. Lenzmann’s work partially supported by National Science Foundation grant DMS-0702492 and a Steno Research Fellowship from the Danish Science Council. Lewin’s work partially supported by Agence Nationale de la Recherche project ACCQUAREL. 257

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