On conditions for transmission in the Sturm-Liouville problem on a network
The authors study various approaches to transmission problems on graphs, on the basis of their reduction to problems for a single differential equation on the graph. For example, just as vibrations of a string can be modelled by a Sturm-Liouville problem on a line segment, vibrations of a woven membrane can be modelled by a Sturm-Liouville problem on a metric graph. The authors show that, for all \(M\) and all sufficiently small \(h\), the natural frequencies less than \(M\) of a uniform woven membrane, with distance \(h\) between adjacent strands, differ from those of the corresponding continuous membrane by less than \(c(M)h^2\), where \(c(M)\) depends on \(M\) but not on \(h\). Other results include a variational principle, theorems on existence and properties of Green's function, and a theorem on differential inequalities on networks, which has as a corollary a network analogue of the Harnack inequality for elliptic problems on manifolds.
scientific article; zbMATH DE number 5000643
scientific article