We consider a semilinear elliptic equation in a varying thin domain of . This thin domain degenerates into a geometric graph when a certain parameter tends to zero. We determine a limit equation on the graph and we prove that a solution of the PDE converges to a solution of the limit equation. Conversely, when a solution of the limit equation is given, we construct a solution of the PDE approaching a solution of the limit equation.
"A semilinear elliptic equation in a thin network-shaped domain." J. Math. Soc. Japan 52 (3) 673 - 697, July, 2000. https://doi.org/10.2969/jmsj/05230673