On certain qualitative properties of equations on a one-dimensional cell complex (Q1390436)

A one-dimensional cell complex \(\overline\Gamma\) is a connected subset in \(\mathbb{R}^3\), composed of a finite quantity of one-dimensional cells \(\sigma_{1i}\) and null-dimensional cells \(\sigma_{0j}\). The differential operator \[ (Lu)(x)= (Du')'(x),\quad x\notin\sigma_{0j},\quad (Lu)(x)=\sum_{x\in\overline{\sigma_{1i}}}D_i(x+){\partial u_i\over\partial\nu} (x),\quad x\in\sigma_{0j}, \] is investigated.