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Wave factorization of elliptic symbols: theory and applications. Introduction to the theory of boundary value problems in non-smooth domains. (English) Zbl 0961.35193
Dordrecht: Kluwer Academic Publishers. ix, 176 p. (2000).
The factorization of the function \(A(\xi),\xi \in \mathbb R^{n}\) with respect to the half-space \(\mathbb R_{+}^{n}=\{ (x'',x_{n})\in \mathbb R^{n}:x_{n}>0\} \), in other words, the representation of \(A(\xi)\) in the form \(A(\xi)=A_{-}(\xi)A_{+}(\xi),\) where \(A_{\pm }(\xi)\) has the analytical continuation with respect to \(\xi _{n}\) on the upper (lower) half-plane, is a very effective tool of solution of boundary value problems for elliptic pseudodifferential equations in the half-space \(\mathbb R_{+}^{n}\) and the investigation of Fredholm properties of elliptic boundary value problems for smooth domains (see, for instance the well-known monography of G. Eskin, AMS, Providence, RI, 1981). If we want to solve the pseudodifferential equation \[ P_{\Gamma }A(D)u_{\Gamma }(x)=f(x), \quad x\in \Gamma , \] in a cone \(\Gamma \) then for an application of the factorization method we need the representation \[ A(\xi)=A_{-\Gamma ^{\ast }}(\xi)A_{\Gamma ^{\ast }}(\xi), \tag{1} \] where \(A_{-\Gamma ^{\ast }}(\xi),A_{\Gamma ^{\ast }}(\xi)\) have analytical continuations in the tube domains \(\mathbb R^{n}+i( -\Gamma ^{\ast }) ,\mathbb R^{n}+i\Gamma ^{\ast },\) respectively, where \(\Gamma ^{\ast }\) is the dual cone \(\Gamma .\) Such an approach to the Riemann problem for tube domains was presented by V. S. Vladimirov in [Izv. Akad. Nauk SSSR, Ser. Mat. 29, 807-834 (1965; Zbl 0166.33704)] and to the Wiener-Hopf equations for the cone \(\Gamma \) by the reviewer in [Teor. Funkts., Funkts. Anal. Prilozh. 5, 59-67 (1967; Zbl 0167.41001)].
But in the book under review first the representation (1) is used systematically for the solution of boundary value problems in domains with angle points and edges. The structure of the book is the following. The first four chapters include material which one needs for reading the following chapters. Chapter 5 is devoted to the basic concept of wave factorization, that is the factorization (1). Chapters 6,7 are devoted to applied problems from diffraction and elasticity theory where the wave factorization method permits to find the solution in a precise form. In Chapter 8 the author considers model pseudodifferential equations in a plane angle. Since a general solution includes arbitrary functions, one needs additional conditions for its unique determination. These conditions are suggested in Chapter 9 as traces of some pseudodifferential operators on angle sides, and then such a boundary value problem is reduced to an equivalent system of integral equations. In Chapter 10 these systems are solved by means of the Mellin transform for some specific cases. Chapter 11 is devoted to classical Dirichlet or Neumann problems for a plane angle. In Chapter 12 problems with co-boundary conditions are considered.

35S15 Boundary value problems for PDEs with pseudodifferential operators
35J40 Boundary value problems for higher-order elliptic equations
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations