Multiple-Scale Analysis of Boundary-Value Problems in Thick Multi-Level Junctions of Type 3:2:2

1. Front Matter

   Pages i-viii

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2. Introduction

   • Taras Mel’nyk, Dmytro Sadovyi

   Pages 1-12

3. Homogenization of Linear Elliptic Problems

   • Taras Mel’nyk, Dmytro Sadovyi

   Pages 13-30

4. Homogenization of Elliptic Problems in Thick Junctions with Sharp Edges

   • Taras Mel’nyk, Dmytro Sadovyi

   Pages 31-50

5. Homogenization of Semilinear Parabolic Problems

   • Taras Mel’nyk, Dmytro Sadovyi

   Pages 51-72

6. Asymptotic Approximations for Solutions to Semilinear Elliptic and Parabolic Problems

   • Taras Mel’nyk, Dmytro Sadovyi

   Pages 73-95

7. Back Matter

   Pages 97-105

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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.

• homogenization
• multiple-scale analysis
• asymptotic approximations of solutions
• boundary value problems
• nonlinear perturbed boundary conditions
• domains with rapidly oscillating boundaries
• multi-level thick junctions
• non-Lipschitz boundaries
• partial differential equations

“The book is well organized and each chapter contains a conclusion with comments and physical interpretation and can be useful to researchers and graduate students interested in asymptotic analysis and applied mathematics.” (Paolo Musolino, zbMATH 1443.35004, 2020)

• Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

  Taras Mel'nyk, Dmytro Sadovyi

Taras A. Mel’nyk is Professor in the Mathematical Physics Department of the Faculty of Mechanics and Mathematics at Taras Shevchenko National University of Kyiv, where he has developed a number of special courses on topics such as asymptotic methods in mathematical physics and the theory of homogenization. A fellow of the Alexander von Humboldt Foundation and member of the American Mathematical Society, he is the author of the textbooks "Complex Analysis" (2015) and "Sobolev Space Theory and Weak Solutions of Boundary Value Problems" (2018). His research interests are related to asymptotic analysis of boundary-value problems, spectral problems, variational inequalities, optimal control problems in domains with complex micro-inhomogeneous structure (perforated materials, composite materials, thick multi-structures, domains with rapidly oscillating boundaries, domains with concentrated masses, thin domains, thin graph-like junctions).

Dmytro Yu. Sadovyi is a postdoctoral researcher at Taras Shevchenko National University of Kyiv, where he obtained his PhD in 2014. His current research interests lie in homogenization of boundary-value problems in thick multi-level junctions.