Effective transmission conditions for second-order elliptic equations on networks in the limit of thin domains

Abstract

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. 2020 Mathematics Subject Classification. 35J15, 35J99, 35B40, 35B25, 49L25, 47H25. Funding. The first author was partially supported by the Air Force Office for Scientific Research grant FA955018-I-0494. The second author was partially supported by the Air Force Office for Scientific Research grant FA9550-18-1-0494, the Office for Naval Research grant N000141712095 and the National Science Foundation grants DMS-1600129 and DMS-1900599. Manuscript received 23rd April 2020, accepted 5th June 2020.