Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian

D. Le Peutrec

doi:10.5802/afst.1265

D. Le Peutrec ¹

¹Laboratoire de Mathématiques, UMR-CNRS 8628, Université Paris-Sud 11, Bâtiment 425, 91405 Orsay, France

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 19 (2010) no. 3-4, pp. 735-809

Abstract (VO)
Résumé

This article follows the previous works [HeKlNi, HeNi] by Helffer-Klein-Nier and Helffer-Nier about the metastability in reversible diffusion processes via a Witten complex approach. Again, exponentially small eigenvalues of some self-adjoint realization of $Δ_{f, h}^{(0)} = - h^{2} Δ + {|\nabla f (x)|}^{2} - h Δ f (x)$ are considered as the small parameter $h > 0$ tends to $0$ . The function $f$ is assumed to be a Morse function on some bounded domain $Ω$ with boundary $\partial Ω$ . Neumann type boundary conditions are considered. With these boundary conditions, some possible simplifications in the Dirichlet problem studied in [HeNi] are no more possible. A finer treatment of the three geometries involved in the boundary problem (boundary, metric, Morse function) is here carried out.

Cet article est dans la continuation des travaux [HeKlNi, HeNi] de Helffer-Klein-Nier et Helffer-Nier sur l’étude de la métastabilité dans des processus de diffusions réversibles via une approche de Witten. Nous considérons encore ici les valeurs propres exponentionnellement petites d’une réalisation auto-adjointe de $Δ_{f, h}^{(0)} = - h^{2} Δ + {|\nabla f (x)|}^{2} - h Δ f (x)$ lorsque le paramètre $h > 0$ tend vers $0$ . La fonction $f$ est une fonction de Morse sur un domaine borné $Ω$ de bord $\partial Ω$ . Des conditions au bord de type Neumann sont considérées ici. Avec ces conditions, certaines simplifications utilisées pour l’étude du problème de Dirichlet dans [HeNi] ne sont plus possibles. Un traitement plus fin des trois géométries intervenant dans le problème à bord (bord, métrique, fonction de Morse) est donc nécessaire.

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Accepté le : 2010-03-29
Publié le : 2011-02-20

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DOI : 10.5802/afst.1265

Affiliations des auteurs :

D. Le Peutrec ¹

¹ Laboratoire de Mathématiques, UMR-CNRS 8628, Université Paris-Sud 11, Bâtiment 425, 91405 Orsay, France

D. Le Peutrec. Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 19 (2010) no. 3-4, pp. 735-809. doi: 10.5802/afst.1265

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