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 are considered as the small parameter tends to . The function is assumed to be a Morse function on some bounded domain with boundary . 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 lorsque le paramètre tend vers . La fonction est une fonction de Morse sur un domaine borné de bord . 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.
@article{AFST_2010_6_19_3-4_735_0, author = {Le Peutrec, D.}, title = {Small eigenvalues of the {Neumann} realization of the semiclassical {Witten} {Laplacian}}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {735--809}, publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques}, address = {Toulouse}, volume = {Ser. 6, 19}, number = {3-4}, year = {2010}, doi = {10.5802/afst.1265}, zbl = {1213.58023}, mrnumber = {2790817}, language = {en}, url = {http://www.numdam.org/articles/10.5802/afst.1265/} }
TY - JOUR AU - Le Peutrec, D. TI - Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2010 SP - 735 EP - 809 VL - 19 IS - 3-4 PB - Université Paul Sabatier, Institut de mathématiques PP - Toulouse UR - http://www.numdam.org/articles/10.5802/afst.1265/ DO - 10.5802/afst.1265 LA - en ID - AFST_2010_6_19_3-4_735_0 ER -
%0 Journal Article %A Le Peutrec, D. %T Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2010 %P 735-809 %V 19 %N 3-4 %I Université Paul Sabatier, Institut de mathématiques %C Toulouse %U http://www.numdam.org/articles/10.5802/afst.1265/ %R 10.5802/afst.1265 %G en %F AFST_2010_6_19_3-4_735_0
Le Peutrec, D. Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. 3-4, pp. 735-809. doi : 10.5802/afst.1265. http://www.numdam.org/articles/10.5802/afst.1265/
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