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Elliptic equations in polyhedral domains. (English) Zbl 1196.35005

Mathematical Surveys and Monographs 162. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4983-5/hbk). vii, 608 p. (2010).
The present book can be considered as the third volume of a trilogy. In the first volume [V. A. Kozlov and the authors, Elliptic boundary value problems in domains with point singularities. Mathematical Surveys and Monographs. 52. Providence, RI: AMS (1997; Zbl 0947.35004)] elliptic BVPs for domains with isolated singularities were considered. The second volume [V. A. Kozlov and the authors, Spectral problems associated with corner singularities of solutions to elliptic equations. Mathematical Surveys and Monographs. 85. Providence, RI: AMS (2001; Zbl 0965.35003)] concerned the corresponding spectra of operator pencils (Mellin operator symbols). The third volume now is devoted to elliptic BVPs with boundary singularities of positive dimension.
The main results of the present book concern pointwise estimates of Green and Poisson kernels, normal solvability in weighted and non weighted Sobolev and Hölder spaces (with corresponding estimates) and Miranda-Agmon type maximum principles. Asymptotic solution representations near the singularities are not in the scope of the book.
The present book consists of three parts. Part I concerns the Dirichlet problem for strongly elliptic systems of arbitrary order. Part II is dedicated to Neumann and mixed problems for elliptic systems of second-order (including the Lamé system). Finally, Part III deals with mixed problems for stationary Stokes and Navier-Stokes systems. More general boundary conditions are not considered in order to state solvability conditions explicitly (without referring to requirements of triviality of kernels and cokernels of auxiliary operators). In each part, the authors pass subsequently from simpler geometric configurations (dihedral cones) to more complicated ones (polyhedral cones and domains diffeomorphic to polyhedrals).
Although the authors work with general settings and aim at optimal results, the presentation is very clear and smooth. All chapters are complemented by many historical and bibliographical remarks and by carefully selected examples.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35J08 Green’s functions for elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35J40 Boundary value problems for higher-order elliptic equations
35J57 Boundary value problems for second-order elliptic systems
35J58 Boundary value problems for higher-order elliptic systems
35A20 Analyticity in context of PDEs
35B50 Maximum principles in context of PDEs
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