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Initial-boundary value and scattering problems in mathematical physics. (English) Zbl 0687.35065
Partial differential equations and calculus of variations, Lect. Notes Math. 1357, 23-60 (1988).
[For the entire collection see Zbl 0648.00008.]
The authors consider initial boundary value problems such as \[ u_ t+Au=0,\quad u(0)=u^ 0, \] where A is a linear differential operator in a three dimensional space. Amongst the problems considered are the spectrum of A, the wave operators and the scattering operator, the behaviour of solutions for large time, the derivation of the scattering operator from the radiation pattern, the asymptotic behaviour of solutions of the Helmholtz equation at high frequencies using the WKBJ ansatz, tangential rays and scattering amplitudes. Finally it is shown how the shape of an obstacle may be derived from a knowledge of the scattering amplitude at high frequencies.
The paper is a survey paper and describes the sort of result which can be obtained, but there is no detailed treatment of any particular problem.
Reviewer: Ll.G.Chambers

MSC:
35P25 Scattering theory for PDEs
35R30 Inverse problems for PDEs
35K15 Initial value problems for second-order parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations