**Mathematical Surveys and Monographs**

Volume: 52;
1997;
414 pp;
Hardcover

MSC: Primary 35;

Print ISBN: 978-0-8218-0754-5

Product Code: SURV/52

List Price: $120.00

Individual Member Price: $96.00

**Electronic ISBN: 978-0-8218-3377-3
Product Code: SURV/52.E**

List Price: $120.00

Individual Member Price: $96.00

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# Elliptic Boundary Value Problems in Domains with Point Singularities

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*V. A. Kozlov; V. G. Maz′ya; J. Rossmann*

This monograph systematically treats a theory of elliptic boundary value problems in domains without singularities and in domains with conical or cuspidal points. This exposition is self-contained and a priori requires only basic knowledge of functional analysis. Restricting to boundary value problems formed by differential operators and avoiding the use of pseudo-differential operators makes the book accessible for a wider readership.

The authors concentrate on fundamental results of the theory: estimates for solutions in different function spaces, the Fredholm property of the operator of the boundary value problem, regularity assertions and asymptotic formulas for the solutions near singular points. A special feature of the book is that the solutions of the boundary value problems are considered in Sobolev spaces of both positive and negative orders. Results of the general theory are illustrated by concrete examples. The book may be used for courses in partial differential equations.

#### Table of Contents

# Table of Contents

## Elliptic Boundary Value Problems in Domains with Point Singularities

- Contents vii8 free
- Introduction 112 free
- Part 1. Elliptic boundary value problems in domains with smooth boundary 718 free
- Chapter 1. Boundary value problems for ordinary differential equations on the half-axis 920
- 1.1. The boundary value problem and its formally adjoint 920
- 1.2. Solvability of the boundary value problem on the half-axis 1324
- 1.3. Solvability of regular problems on the half-axis in Sobolev spaces of negative order 2031
- 1.4. Properties of the operator adjoint to the operator of the boundary value problem 2637

- Chapter 2. Elliptic boundary value problems in the half-space 3142
- Chapter 3. Elliptic boundary value problems in smooth domains 5970
- 3.1. The boundary value problem and its formally adjoint 5970
- 3.2. An a priori estimate for the solution 7283
- 3.3. The adjoint operator 8091
- 3.4. Solvability of elliptic boundary value problems in smooth domains 8495
- 3.5. The Green function of the boundary value problem 90101
- 3.6. Elliptic boundary value problems with parameter 98109

- Chapter 4. Variants and extensions 105116

- Part 2. Elliptic problems in domains with conical points 143154
- Chapter 5. Elliptic boundary value problems in an infinite cylinder 145156
- 5.1. Operator-valued polynomials and applications to ordinary differential equations with operator coefficients 145156
- 5.2. Solvability of the model problem in an infinite cylinder 153164
- 5.3. Solvability of the model problem in the cylinder in Sobolev spaces of negative order 161172
- 5.4. Asymptotics of the solution of the model problem at infinity 168179
- 5.5. The boundary value problem with coefficients which stabilize at infinity 180191

- Chapter 6. Elliptic boundary value problems in domains with conical points 191202
- 6.1. The model problem in an infinite cone 191202
- 6.2. Elliptic boundary value problems in a bounded domain with conical points 212223
- 6.3. Solvability of elliptic boundary value problems in bounded domains with conical points 219230
- 6.4. Asymptotics of the solution 234245
- 6.5. Boundary value problems with parameter in domains with conical points 246257
- 6.6. Examples 260271

- Chapter 7. Elliptic boundary value problems in weighted Sobolev spaces with nonhomogeneous norms 267278
- Chapter 8. Variants and extensions 303314

- Part 3. Elliptic problems in domains with cuspidal points 335346
- Bibliography 397408
- Index 409420
- List of Symbols 413424

#### Readership

Graduate students and research mathematicians interested in partial differential equations.

#### Reviews

In the present book attention is focused on boundary value problems in regions in which the boundary has a finite number of point singularities … The behavior of the solution to an elliptic boundary value problem in a domain with singularities is of its nature complicated. The book develops the theory in a clear form. Careful treatment is given to formulas for the coefficients in the expansion into singular functions, the so-called stress intensity factors. There is a careful treatment of the adjoint problem and of data which are distributions. The book contains a lot of material and will be a valuable resource for researchers in mechanics and fluid dynamics and for numerical analysts concerned with the solution of these problems.

-- SIAM Review

The authors have obtained many deep results for elliptic boundary value problems in domains with singularities … without doubt, the book will be very interesting for many mathematicians working with elliptic boundary problems in smooth and nonsmooth domains, and it would be frequently used in any mathematical library.

-- Mathematical Reviews

The book is a welcome addition, which will, we hope, expose a wider mathematical audience (in particular, applied mathematicians), to these results. The book can be recommended to specialists in partial differential equations as an accessible and up-to-date research monograph. At the same time, it is a good text for graduate students specialising in the subject.

-- Bulletin of the London Mathematical Society

Very well written with a clear and concise style of exposition and elegant proofs. This fact together with the deep and profound content and the excellent choice of material make it a distinguished and valuable reading for researchers and graduate students in the field of partial differential equations.

-- Zentralblatt MATH