This article aims to continue the study of geometric optics expansions for hyperbolic boundary value problems in the quarter-space initiated in [2]. The motivations are linked to the range of effective applicability of the theorem establishing the existence of the geometric optics expansions. Compared to [2], we ameliorate the range of applicability by adding two distinct features. The first one is that now we can consider glancing modes in the expansions by using the results of [17]. The second one, which is proper to quarter-space problems, is that we can now consider rather “complicated” self-interaction phenomena. It is a first step in the study of geometric optics expansions in bounded domains. A direct consequence of the first point of amelioration is that no condition on glancing modes is required to intialize the construction of the geometric optics expansion. It seems to indicate that the expected condition characterizing the strong well-posedness of corner problems, established in [14], can be relaxed to the hyperbolic component of the stable subspace only.
Cet article vise à poursuivre l’étude des développements d’optique géométrique pour les problèmes aux limites hyperboliques posés dans un quart d’espace, étude initiée dans [2]. Les motivations sont ici liées au domaine d’applicabilité effective du théorème établissant l’existence de tels développements. Comparé à [2], nous avons amélioré le domaine d’applicabilité de deux façons distinctes. D’abord, nous pouvons maintenant considérer dans les développements des modes rasants en adaptant les résultats de [17]. Ensuite, ceci est propre à la géométrie du quart d’espace, nous pouvons maintenant considérer des phénomènes d’auto-interaction assez « complexes ». Ceci constitue une première étape nécessaire dans la construction de développements d’optique géométrique dans des géométries bornées. Une conséquence directe de notre nouvelle contribution est que, pour son initialisation, la résolution de la cascade d’équations ne nécessite pas de condition sur les modes rasants. Ceci semble indiquer que la condition que l’on croit caractériser les problèmes fortement bien-posés de [14] pourrait être relaxée sur les modes hyperboliques seulement.
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Keywords: hyperbolic boundary value problem, corner problem, geometric optics expansion, self-interaction, glancing modes
Antoine Benoit 1
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@article{AFST_2025_6_34_2_413_0,
author = {Antoine Benoit},
title = {Geometric optics expansions for quarter-space boundary value problems {III:} glancing modes and multiple self-interaction},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {413--499},
publisher = {Universit\'e de Toulouse, Toulouse},
volume = {Ser. 6, 34},
number = {2},
year = {2025},
doi = {10.5802/afst.1817},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1817/}
}
TY - JOUR AU - Antoine Benoit TI - Geometric optics expansions for quarter-space boundary value problems III: glancing modes and multiple self-interaction JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2025 SP - 413 EP - 499 VL - 34 IS - 2 PB - Université de Toulouse, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1817/ DO - 10.5802/afst.1817 LA - en ID - AFST_2025_6_34_2_413_0 ER -
%0 Journal Article %A Antoine Benoit %T Geometric optics expansions for quarter-space boundary value problems III: glancing modes and multiple self-interaction %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2025 %P 413-499 %V 34 %N 2 %I Université de Toulouse, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1817/ %R 10.5802/afst.1817 %G en %F AFST_2025_6_34_2_413_0
Antoine Benoit. Geometric optics expansions for quarter-space boundary value problems III: glancing modes and multiple self-interaction. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 34 (2025) no. 2, pp. 413-499. doi : 10.5802/afst.1817. https://afst.centre-mersenne.org/articles/10.5802/afst.1817/
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