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Polygonal Interface Problems


Serge Nicaise

The aim of this volume is to extend some of the classical results of Kondratiev, Maz'ya-Plamenevskii, Grisvard and Dauge, about the singular behaviour of the weak solution of a boundary value problem in a nonsmooth domain to the setting of boundary value problems on a two-dimensional polygonal topological network (roughly speaking it is a network such that each face is a polygon). Some mechanical phenomena were recently described by boundary value problems on such networks: let us quote vibrating folded membranes, composite plates, folded plates, junctions in elastic multi-structures. In view to numerical applications, the singularities, as well as their coefficients (the so-called stress intensity factors) are described explicitly. The obtained decomposition into regular and singular parts also leads to the Fredholm property of the associated operator between appropriate Sobolev spaces.
Contents: Sobolev spaces - Networks - Boundary value problems - Interface (transmission) problems - Fredholm operators - Singularities - Coefficients of the singularities - Differential equatious in Hilbert spaces.