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.
Accepted:
Published online:
Pierre-Louis Lions 1, 2; Panagiotis E. Souganidis 3
@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}, }
TY - JOUR AU - Pierre-Louis Lions AU - Panagiotis E. Souganidis 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 DO - 10.5802/crmath.83 LA - en ID - CRMATH_2020__358_7_797_0 ER -
%0 Journal Article %A Pierre-Louis Lions %A Panagiotis E. Souganidis %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 %R 10.5802/crmath.83 %G en %F CRMATH_2020__358_7_797_0
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, Volume 358 (2020) no. 7, pp. 797-809. doi : 10.5802/crmath.83. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.83/
[1] Hamilton–Jacobi equations on networks as limits of singularly perturbed problems in optimal control: dimension reduction, Commun. Partial Differ. Equations, Volume 40 (2015) no. 4, pp. 652-693 | DOI | MR | Zbl
[2] Ergodic problems and periodic homogenization for fully nonlinear equations in half-space type domains with Neumann boundary conditions, Indiana Univ. Math. J., Volume 57 (2008) no. 5, pp. 2355-2375 | DOI | MR | Zbl
[3] Fully nonlinear elliptic equations, Colloquium Publications, 43, American Mathematical Society, 1995 | MR | Zbl
[4] User’s guide to viscosity solutions of second order partial differential equations, Bull. Am. Math. Soc., Volume 27 (1992) no. 1, pp. 1-67 | DOI | MR | Zbl
[5] A PDE approach to small stochastic perturbations of Hamiltonian flows, J. Differ. Equations, Volume 252 (2012) no. 2, pp. 1748-1775 | DOI | MR | Zbl
[6] Well-posedness for multi-dimensional junction problems with Kirchoff-type conditions, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl., Volume 28 (2017) no. 4, pp. 807-816 | DOI | MR | Zbl
[7] Comparison principles and pointwise estimates for viscosity solutions, Rev. Mat. Iberoam., Volume 4 (1988) no. 3-4, pp. 453-468 | DOI | Zbl
Cited by Sources:
Comments - Policy