معادلات بیضوی در دامنه‌های چندوجهی (بررسی‌ها و مونوگرافی‌های ریاضی)

This is the first monograph which systematically treats elliptic boundary value problems in domains of polyhedral type. The authors mainly describe their own recent results focusing on the Dirichlet problem for linear strongly elliptic systems of arbitrary order, Neumann and mixed boundary value problems for second order systems, and on boundary value problems for the stationary Stokes and Navier-Stokes systems. A feature of the book is the systematic use of Green's matrices. Using estimates for the elements of these matrices, the authors obtain solvability and regularity theorems for the solutions in weighted and non-weighted Sobolev and Holder spaces. Some classical problems of mathematical physics (Laplace and biharmonic equations, Lame system) are considered as examples. Furthermore, the book contains maximum modulus estimates for the solutions and their derivatives. The exposition is self-contained, and an introductory chapter provides background material on the theory of elliptic boundary value problems in domains with smooth boundaries and in domains with conical points. The book is destined for graduate students and researchers working in elliptic partial differential equations and applications Contents 6 Introduction 10 Part 1. The Dirichlet problem for strongly elliptic systems in polyhedral domains 16 Chapter 1. Prerequisites on elliptic boundary value problems in domains with conical points 18 Chapter 2. The Dirichlet problem for strongly elliptic systems in a dihedron 32 Chapter 3. The Dirichlet problem for strongly elliptic systems in a cone with edges 98 Chapter 4. The Dirichlet problem in a bounded domain of polyhedral type 150 Chapter 5. The Miranda-Agmon maximum principle 170 Part 2. Neumann and mixed boundary value problems for second order systems in polyhedral domains 220 Chapter 6. Boundary value problems for second order systems in a dihedron 222 Chapter 7. Boundary value problems for second order equations in a polyhedral cone 298 Chapter 8. Boundary value problems for second order systems in a bounded polyhedral domain 364 Part 3. Mixed boundary value problems for stationary Stokes and Navier-Stokes systems in polyhedral domains 388 Chapter 9. Boundary value problem for the Stokes system in a dihedron 390 Chapter 10. Mixed boundary value problems for the Stokes system in a polyhedral cone 452 Chapter 11. Mixed boundary value problems for the Stokes and Navier-Stokes systems in a bounded domain of polyhedral type 528 Historical remarks 590 Bibliography 598 List of Symbols 608 List of Examples 614 Index 616 This is the first monograph which systematically treats elliptic boundary value problems in domains of polyhedral type. The authors mainly describe their own recent results focusing on the Dirichlet problem for linear strongly elliptic systems of arbitrary order, Neumann and mixed boundary value problems for second order systems, and on boundary value problems for the stationary Stokes and Navier–Stokes systems. A feature of the book is the systematic use of Green's matrices. Using estimates for the elements of these matrices, the authors obtain solvability and regularity theorems for the solutions in weighted and non-weighted Sobolev and Hölder spaces. Some classical problems of mathematical physics (Laplace and biharmonic equations, Lamé system) are considered as examples. Furthermore, the book contains maximum modulus estimates for the solutions and their derivatives. The exposition is self-contained, and an introductory chapter provides background material on the theory of elliptic boundary value problems in domains with smooth boundaries and in domains with conical points. The book is destined for graduate students and researchers working in elliptic partial differential equations and applications. Pt. 1. The Dirichlet Problem For Strongly Elliptic Systems In Polyhedral Domains -- 2. Neumann And Mixed Boundary Value Problems For Second Order Systems In Polyhedral Domains -- Pt. 3. Mixed Boundary Value Problems For Stationary Stokes And Navier-stokes Systems In Polyhedral Domains. Vladimir Maz'ya, Jürgen Rossmann. Includes Bibliographical References (p. 589-597) And Index.