**Mathematical Surveys and Monographs**

Volume: 162;
2010;
608 pp;
Hardcover

MSC: Primary 35;

Print ISBN: 978-0-8218-4983-5

Product Code: SURV/162

List Price: $123.00

Individual Member Price: $98.40

**Electronic ISBN: 978-1-4704-1389-7
Product Code: SURV/162.E**

List Price: $123.00

Individual Member Price: $98.40

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#### Supplemental Materials

# Elliptic Equations in Polyhedral Domains

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*Vladimir Maz′ya; Jürgen Rossmann*

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.

#### Table of Contents

# Table of Contents

## Elliptic Equations in Polyhedral Domains

- Contents v6 free
- Introduction 110 free
- Part 1. The Dirichlet problem for strongly elliptic systems in polyhedral domains 716 free
- Chapter 1. Prerequisites on elliptic boundary value problems in domains with conical points 918
- Chapter 2. The Dirichlet problem for strongly elliptic systems in a dihedron 2332
- Chapter 3. The Dirichlet problem for strongly elliptic systems in a cone with edges 8998
- Chapter 4. The Dirichlet problem in a bounded domain of polyhedral type 141150
- Chapter 5. The Miranda-Agmon maximum principle 161170
- Part 2. Neumann and mixed boundary value problems for second order systems in polyhedral domains 211220
- Chapter 6. Boundary value problems for second order systems in a dihedron 213222
- Chapter 7. Boundary value problems for second order equations in a polyhedral cone 289298
- Chapter 8. Boundary value problems for second order systems in a bounded polyhedral domain 355364
- Part 3. Mixed boundary value problems for stationary Stokes and Navier-Stokes systems in polyhedral domains 379388
- Chapter 9. Boundary value problem for the Stokes system in a dihedron 381390
- Chapter 10. Mixed boundary value problems for the Stokes system in a polyhedral cone 443452
- Chapter 11. Mixed boundary value problems for the Stokes and Navier-Stokes systems in a bounded domain of polyhedral type 519528
- Historical remarks 581590
- Bibliography 589598
- List of Symbols 599608
- List of Examples 605614
- Index 607616 free

#### Readership

Graduate students and research mathematicians interested in elliptic PDEs.

#### Reviews

The book...is the third book in a series of books on the subject by these well-known and prolific authors. [It] makes...welcome additions to the previous works of these authors. [The] results [are] useful in applications, such as...obtaining optimal rates of convergence of the finite element method.

-- Victor Nistor, Mathematical Reviews