no. 7
Équations aux dérivées partielles
Effective transmission conditions for second-order elliptic equations on networks in the limit of thin domains
Lions, Pierre-Louis1, 2  ; Souganidis, Panagiotis E.3
1 Collège de France, 11 Place Marcelin Berthelot, 75005 Paris, France
2 CEREMADE, Université de Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75016 Paris, France
3 Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USA
Comptes Rendus. Mathématique, Tome 358 (2020) no. 7, pp. 797-809.

We consider star-shaped tubular domains consisting of a number of non intersecting semi-infinite strips of small thickness that are connected by a central region of diameter proportional to the thickness of the strips. At the thin-domain limit, the region reduces to a network of half-lines with the same end point (junction). We show that the solutions of uniformly elliptic partial differential equations set on the domain with Neumann boundary conditions converge, in the thin-domain limit, to the unique solution of a second-order partial differential equation on the network satisfying an effective Kirchhoff-type transmission condition at the junction. The latter is found by solving an “ergodic”-type problem at infinity obtained after a first-order blow up at the junction.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.83
Classification : 35J15, 35J99, 35B40, 35B25, 49L25, 47H25
Lions, Pierre-Louis 1, 2 ; Souganidis, Panagiotis E. 3

1 Collège de France, 11 Place Marcelin Berthelot, 75005 Paris, France
2 CEREMADE, Université de Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75016 Paris, France
3 Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USA 
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     title = {Effective transmission conditions for second-order elliptic equations on networks in the limit of thin domains},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {797--809},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
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     language = {en},
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Lions, Pierre-Louis; Souganidis, Panagiotis E. Effective transmission conditions for second-order elliptic equations on networks in the limit of thin domains. Comptes Rendus. Mathématique, Tome 358 (2020) no. 7, pp. 797-809. doi : 10.5802/crmath.83. http://www.numdam.org/articles/10.5802/crmath.83/

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