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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 (ISBN 2-225-82543-2). x, 198 p. (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.

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.

Reviewer: István Ecsedi (Miskolc-Egyetemváros)

### 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.) |