A semilinear elliptic equation in a thin network-shaped domain

Abstract

We consider a semilinear elliptic equation in a varying thin domain of $R^n$. 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.