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Mathematical Surveys and Monographs
1997; 414 pp; hardcover
List Price: US$120
Member Price: US$96
Order Code: SURV/52
Elliptic Equations in Polyhedral Domains - Vladimir Maz'ya and Jurgen 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.
Graduate students and research mathematicians interested in partial differential equations.
"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
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