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.
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@article{CRMATH_2020__358_7_797_0,
author = {Lions, Pierre-Louis and Souganidis, Panagiotis E.},
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},
number = {7},
year = {2020},
doi = {10.5802/crmath.83},
language = {en},
url = {http://www.numdam.org/articles/10.5802/crmath.83/}
}
TY - JOUR AU - Lions, Pierre-Louis AU - Souganidis, Panagiotis E. TI - Effective transmission conditions for second-order elliptic equations on networks in the limit of thin domains JO - Comptes Rendus. Mathématique PY - 2020 SP - 797 EP - 809 VL - 358 IS - 7 PB - Académie des sciences, Paris UR - http://www.numdam.org/articles/10.5802/crmath.83/ DO - 10.5802/crmath.83 LA - en ID - CRMATH_2020__358_7_797_0 ER -
%0 Journal Article %A Lions, Pierre-Louis %A Souganidis, Panagiotis E. %T Effective transmission conditions for second-order elliptic equations on networks in the limit of thin domains %J Comptes Rendus. Mathématique %D 2020 %P 797-809 %V 358 %N 7 %I Académie des sciences, Paris %U http://www.numdam.org/articles/10.5802/crmath.83/ %R 10.5802/crmath.83 %G en %F CRMATH_2020__358_7_797_0
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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