The aim of this book is to extend some of the' classical results of Kondratiev
[60], Maz'ya-Plamenevskii [78, 79], Grisvard [49] and Dauge [33], concern-
ing the singular behaviour of the weak solution of a boundary value prob-
lem (written in the sequel b.v.p.) in a nonsmooth domain to the setting of
b. v .p.'s on a two-dimensional polygonal topological network (roughly speak-
ing it is a network such that each face is a polygon; for the sake of briefness
we shall note 2-d. network). Indeed some physical phenolnena were recently
described by b. v. p.'s on such networks (or even on more cOll1plicated ones):
let us quote vibrating folded membranes [65], composite plates [39], folded
plates [64, 65], junctions in elastic multi-structures [26] for instance. Here
for the sake of simplicity, we treat some particular b.v.p.'s but it is clear
that we work out the appropriate background which allows to solve a large
class of b.v.p.'s on 2-d. networks.