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Polygonal interface problems. (English) Zbl 0794.35040
Methoden und Verfahren der Mathematischen Physik. 39. Frankfurt/Main: Peter Lang. 250 p. (1993).
The purpose of this book is to present some classical results of the theory of boundary value problems for elliptic equations in nonsmooth domains in the framework of boundary value problems on two dimensional polygonal topological networks. Here is the precise definition of a network: \(\Omega\) is a two dimensional polygonal topological network iff \(\Omega\subset\mathbb{R}^ n\) \((n\) fixed \(\geq 2)\) is formed by a finite union of disjoint nonempty subsets \(P_ i\), \(i\in{\mathcal I}\), such that
(i) each \(P_ i\) is a simply connected open subset of a plane \(\Pi_ i\subset\mathbb{R}^ n\), \(P_ i\) being a polygonal domain of \(\Pi_ i\); (ii) \(\bigcup_{i\in{\mathcal I}}\overline P_ i\) is connected; (iii) \(\forall i\), \(j\in{\mathcal I}\), \(i\neq j\), \(\overline P_ i\cap\overline P_ j\) is either empty or a common vertex or a whole common side.
The book is divided into six chapters. The first chapter is dedicated to a review of well-known results on Sobolev spaces, Green formulas and a priori estimates for elliptic operators. Chapters 2 and 3 are dedicated to elliptic transmission problems and their two-dimensional network counterpart. Chapters 4 and 5 review the (abstract) theory of the boundary value problems in nonsmooth domains in the framework of Hilbert spaces and the associated a priori estimates in weighted Sobolev spaces; these results are then established in the network context. Finally the 6th chapter is devoted to a pair of coupled problems in a polygonal domain, notably the Laplace operator in a part of the considered domain and the biharmonic operator on the other part. The core of the book consists of chapters 5 and 6.
A major reference as well as a primary source book for the present book is P. Grisvard’s, elliptic problems in nonsmooth domains (Monographs and Studies in Mathematics 24, Pitman, Boston, 1985; Zbl 0695.35060).
Reviewer: A.Bove (Bologna)

35J25 Boundary value problems for second-order elliptic equations
35J40 Boundary value problems for higher-order elliptic equations
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35B45 A priori estimates in context of PDEs
35C10 Series solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs