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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.

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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DOI : https://doi.org/10.5802/crmath.83
Classification : 35J15,  35J99,  35B40,  35B25,  49L25,  47H25
@article{CRMATH_2020__358_7_797_0,
     author = {Pierre-Louis Lions and Panagiotis E. Souganidis},
     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},
}
Pierre-Louis Lions; Panagiotis E. Souganidis. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.83/

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