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Elliptic problems in nonsmooth domains. Reprint of the 1985 hardback ed. (English) Zbl 1231.35002

Classics in Applied Mathematics 69. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 978-1-611972-02-3/pbk). xx, 410 p. (2011).
The book is a reprint of the publication reviewed in [(1985; Zbl 0695.35060)].
The monograph concerns the comprehensive treatment of elliptic boundary value problems in domains with nonsmooth boundaries and problems with mixed boundary conditions. The main topic of the book is the transformation of the so-called shift theorem from smooth boundary value problems into various boundary value problems where boundaries are no longer smooth.
The problems presented in the book are formulated on domains which can arise in practice, in particular those whose geometry is simple but not smooth, such as three-dimensional polyhedra or two-dimensional reductions which are more convenient for the numerical treatment. For this purpose the book is mainly intended for mathematicians working in the fields of elliptic differential equations and numerical analysis as well as for those working with applications of boundary value problems.
The book is organized as follows. Chapter 1 is devoted to Sobolev spaces. The author presents results on the properties of Sobolev spaces, in particular the boundary properties of functions belonging to Sobolev spaces on domains with polygonal boundaries. Chapter 2 is devoted to regular second-order elliptic boundary value problems in which the boundary has \(C^{1,1}\)-regularity. The reader finds here existence, uniqueness and smoothness results, a priori estimates as well as applications of the Fredholm theory and maximum principle. In Chapter 3, second-order elliptic boundary value problems in convex domains are considered. For such domains the a priori estimates can be improved and, consequently, the regularity results can be strengthened. In Chapter 4 the author proves a modified shift theorem for general second-order elliptic boundary value problems in a plane polygon. On each side of the polygon either a Dirichlet, a Neumann, or an oblique boundary condition applies. This chapter contains the proof of the regularity of the second derivatives of the solution, while in Chapter 5 we find the regularity of the higher derivatives. These chapters also contain problems involving non-homogeneous operators as well as operators with variable coefficients. Chapter 6 contains the Hölder space approach to the considered problems. We find here the Schauder inequalities in both regular second-order elliptic boundary value problems as well as in the polygon case. Chapter 7 focuses on the Dirichlet problem for the biharmonic equation in a plane polygon. A suitably modified shift theorem for this case is proved and reformulated for the linear Stokes system and the stationary Navier-Stokes equations in a plane polygon. Chapter 8 contains miscellaneous related topics. The Bibliography contains a rich set of relevant references.

MathOverflow Questions:

A property of Lipschitz domains

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35B65 Smoothness and regularity of solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47F05 General theory of partial differential operators

Citations:

Zbl 0695.35060
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