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## Über dieses Buch

This book presents asymptotic methods for boundary-value problems (linear and semilinear, elliptic and parabolic) in so-called thick multi-level junctions. These complicated structures appear in a large variety of applications.

A concise and readable introduction to the topic, the book provides a full review of the literature as well as a presentation of results of the authors, including the homogenization of boundary-value problems in thick multi-level junctions with non-Lipschitz boundaries, and the construction of approximations for solutions to semilinear problems.

Including end-of-chapter conclusions discussing the results and their physical interpretations, this book will be of interest to researchers and graduate students in asymptotic analysis and applied mathematics as well as to physicists, chemists and engineers interested in processes such as heat and mass transfer.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
A full literature review of the homogenization of various boundary-value problems in thick junctions is presented and different asymptotic methods used in those works are described, so that interested researchers could quickly read and understand the topic. We also explain the choice of boundary-value problems in thick multilevel junctions of type 3:2:2.

### Chapter 2. Homogenization of Linear Elliptic Problems

Abstract
Convergence theorems are proved for solutions to linear elliptic boundary-value problems in a thick multilevel junction of type 3:2:2. In the first problem, various alternating perturbed Robin boundary conditions are considered, and alternating Neumann and Dirichlet boundary conditions on the surfaces of the thin discs from different sets in the second one. The convergence of the energy integrals for each problem is also proved (this is a very useful result that gives the possibility to directly obtain results for optimal control problems involving thick multilevel junctions).

### Chapter 3. Homogenization of Elliptic Problems in Thick Junctions with Sharp Edges

Abstract
We consider the case when the thin discs of a thick multilevel junction can have sharp edges, i.e., their thickness tends to zero polynomially while approaching the edges. Three qualitatively different cases in the asymptotic behavior of the solution to a linear elliptic boundary-value problem are discovered depending on the edge form, namely rounded edges; linear wedges; and very sharp edges (in this case the boundary is not Lipshitz). Nonstandard and new techniques are proposed to get the corresponding homogenized problems (untypical in the last two cases). The obtained results mathematically justify an interesting physical effect for heat radiators (see conclusions to this chapter).

### Chapter 4. Homogenization of Semilinear Parabolic Problems

Abstract
The method proposed in Chap. 2 is broadened for semilinear initial-boundary-value problems. Here we show how to apply the Minty–Browder method to homogenize nonlinear Robin conditions that have special intensity factor $$\varepsilon ^\alpha$$; where a is a real parameter that significantly impacts the asymptotic behavior of the solutions.

### Chapter 5. Asymptotic Approximations for Solutions to Semilinear Elliptic and Parabolic Problems

Abstract
Approximation techniques are demonstrated for semilinear elliptic and parabolic problems with various alternating Robin boundary conditions. With the help of special junction-layer solutions, whose behavior is determined by the type 3:2:2, and the method of matched asymptotic expansions, approximation functions are constructed for the solutions and the corresponding asymptotic estimates in Sobolev spaces are proved. These estimates show the influence of the given data and parameters on the asymptotic behavior of the solutions.