Continuity and discontinuity  of the boundary layer tail

Feldman, William M.; Kim, Inwon C.

doi:10.24033/asens.2338

Geodesic

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Continuity and discontinuity of the boundary layer tail
[Continuité et discontinuité de la queue de la couche limite]

Feldman, William M. ; Kim, Inwon C.

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 4, pp. 1017-1064.

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Abstract (VO)
Résumé

We investigate the continuity properties of the homogenized boundary data $\bar{g}$ for oscillating Dirichlet boundary data problems. The homogenized boundary condition arises as the boundary layer tail of a problem set in a half-space. The continuity properties of this boundary layer tail depending on the normal direction of the half space play an important role in the homogenization process in general bounded domains. We show that, for a generic non-rotation-invariant operator and boundary data, $\bar{g}$ is discontinuous at every rational direction. In particular this implies that the continuity condition of Choi and Kim [16] is essentially sharp. On the other hand, when the condition of [16] holds, we show a Hölder modulus of continuity for $\bar{g}$ . When the operator is linear we show that $\bar{g}$ is Hölder- $\frac{1}{d}$ up to a logarithmic factor. The proofs are based on a new geometric observation on the limiting behavior of $\bar{g}$ at rational directions, reducing to a class of two dimensional problems for projections of the homogenized operator.

Nous étudions les propriétés de continuité des données sur les bords homogénéisées $\bar{g}$ pour des problèmes de Dirichlet avec des données oscillantes. La condition au bord homogénéisée se pose comme la queue de la couche limite d'un problème posé dans un demi-espace. Les propriétés de cette queue de la couche limite en fonction de la direction normale du demi-espace jouent un rôle important dans le processus d'homogénéisation dans des domaines bornés généraux. Nous montrons que, pour un opérateur non-rotation invariant générique et les données au bord, $\bar{g}$ est discontinu à chaque direction rationnelle. En particulier, cela implique que la condition de continuité de Choi et Kim [16] est essentiellement sharp. D'autre part, lorsque la condition de [16] est satisfaite, nous montrons un module de continuité Hölder pour $\bar{g}$ . Lorsque l'opérateur est linéaire, nous montrons que $\bar{g}$ est $\frac{1}{d} -$ Hölder jusqu'à un facteur logarithmique. Les preuves sont basées sur une nouvelle observation géométrique sur le comportement limite de $\bar{g}$ dans des directions rationnelles, ce qui réduit à une classe de problèmes deux dimensionnelles pour les projections de l'opérateur homogénéisé.

Publié le : 2017-05-27

MR Zbl

DOI : 10.24033/asens.2338

Classification : 35J60, 35J57, 35B27, 76F40.
Keywords: Homogenization, oscillating boundary data, fully nonlinear elliptic equations, boundary layers.
Mots-clés : Homogénéisation, données oscillantes sur les bords, équation elliptique complément non linéaire, couche limite.

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     author = {Feldman, William M. and Kim, Inwon C.},
     title = {Continuity and discontinuity  of the boundary layer tail},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1017--1064},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 50},
     number = {4},
     year = {2017},
     doi = {10.24033/asens.2338},
     mrnumber = {3679620},
     zbl = {1381.35039},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.24033/asens.2338/}
}

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Feldman, William M.; Kim, Inwon C. Continuity and discontinuity  of the boundary layer tail. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 4, pp. 1017-1064. doi : 10.24033/asens.2338. http://geodesic.mathdoc.fr/articles/10.24033/asens.2338/

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