Differential equations on networks (geometric graphs)
This paper is a short survey on the problems and trends of the recently developped theory of linear boundary value problems on graphs and networks. It contains an introduction to the physical models that motivate the theory, e.g., grids of strings, electrical networks. The main concepts are introduced, starting with the notion of a geometric graph (one-dimensional manifold where we can single out edges and vertices). A linear boundary value problem (bvp) of order \(n\) on a graph consists of a set of linear differential equations of order \(n\), one at each edge, together with a set of linear consistence conditions at each vertex (these may include boundary conditions) involving the derivatives of orders \(\leq n-1\). As an example, the Dirichlet problem for a 2nd-order Sturm-Liouville problem is introduced. In reviewing the results achieved so far, the authors describe the scope of a substantial part of the literature on this kind of problems and particularly their contributions on the maximum principle, Green's functions and fourth-order bvps. A few examples are then discussed in some detail in order to give the reader a flavour of the kind of problems at stake. We mention the following topics: 1) Spectral theory for Sturm-Liouville bvps on graphs may deviate a lot from the classical one. If one considers a pair of equal strings placed along the axes and tied at a common center at the origin, we have to analyse the bvp \[ -u_i''=\lambda u_i, \quad u_i(\pm\pi)=0,\;i=1,2,\quad u_1(0)=u_2(0), \] \[ u'_1(0^+)-u'_1(0^-)+u'_2(0^+)-u'_2(0^-)=0. \] It turns out that there are eigenvalues with multiplicity equal to 3. This feature appears even if the system loses symmetry. 2) By ignoring one of the edges in the preceeding cross one obtains a ``star'' composed of three strings. With a further simplification (no mass at one of the strings), the problem to be addressed becomes \[ -u''=\lambda u,\;-l_1\leq x\leq l_2,\quad u(0^-)=u(0^+)=l_3(u'(0^+)-u'(0^-)), \] one interpretation of which leads to a single equation with a distributional coefficient \(-u''+\frac{1}{l_3}\delta(x)u=\lambda u\). This example shows that oscillation theory and properties of the zeros of eigenfunctions cannot be established. 3) The model for a square membrane with fixed boundary versus a bvp on a string network that may be seen as a discretization of the former. The equations for eigenvalues of the bvp on the network are obtained and it is shown that they are approximations to the spectrum of the membrane. 4) The study of Green's function for \[ -u''=f(x), \;x\neq\xi,\quad u'(\xi^+)-u'(\xi^-)=f(\xi),\quad u(0)=0=u(1),\quad u(\xi^+)=u(\xi^-), \] and its relation with Green's function for the classical elementary Dirichlet problem. The paper is very well written and a good guide for the literature.
boundary value problems in graphs and networks
scientific article; zbMATH DE number 2174267