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Variational problems in multi-domains. Modeling of junctions and applications. (Problèmes variationnels dans les multi-domaines. Modélisation des jonctions et applications.) (French) Zbl 0744.73027
Recheres en Mathématiques Appliquées. 19. Paris etc.: Masson. x, 198 pp. (1991).
This book is a successful attempt to explain an important new mathematical method to applied mathematicians and engineers interested in research and recent developments in the modelling of multi-structures. The multi-structures can comprise substructures and different dimensions of their components (three dimensional structures, plates, beams). The coupled and multidimensional boundary value problems of heat conduction and elasticity are considered. Asymptotic expansion methods and variational formulations are used to derive the essential equations.
Chapter 1 deals with the fundamental results of functional analysis (distributions, Sobolev spaces, variational problems). These mathematical backgrounds have been mainly used in the next chapters. In Chapter 2 junctions between slender structures in the statical case are treated in connection with the Laplace operator. The heat equation in $$L$$-shaped domains and three dimensional – two-dimensional domains in the dynamical case and the wave equation in $$L$$-shaped domains are considered in Chapter 3. The Chapter 4 deals with the junctions of elastic plates in detailed form. The statical and dynamical problems of multi-plate structures are studied by asymptotic analysis. The multi-rod structures and some generalizations are presented in the Chapter 5.
The main results of this book are as follows: (a) a general technique for modelling of junction; (b) the reducing of three (two)-dimensional problems to two (one)-dimensional problems.

##### MSC:
 74K20 Plates 49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control 74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids 74S30 Other numerical methods in solid mechanics (MSC2010) 74P10 Optimization of other properties in solid mechanics 74E30 Composite and mixture properties 35C20 Asymptotic expansions of solutions to PDEs 74K10 Rods (beams, columns, shafts, arches, rings, etc.)