1st Edition

Asymptotic Methods in Equations of Mathematical Physics

By B Vainberg Copyright 1989
    506 Pages
    by CRC Press

    This book provides a single source for both students and advanced researchers on asymptotic methods employed in the linear problems of mathematical physics. It opens with a section based on material from special courses given by the author, which gives detailed coverage of classical material on the equations of mathematical physics and their applications, and includes a simple explanation of the Maslov Canonical Operator method. The book goes on to present more advanced material from the author's own research. Topics range from radiation conditions and the principle of limiting absorption for general exterior problems, to complete asymptotic expansion of spectral function of equations over all of space. This book serves both as a manual and teaching aid for students of mathematics and physics and, in summarizing for the first time in a monograph problems previously investigated in journal articles, as a comprehensive reference for advanced researchers.

    art 1 The stationary phase method: on asymptotic expansions; the stationary phase method ; the stationary phase method, the multidimensional case; the problem of waves on the surface of a liquid; the asymptotic behaviour of the Fourier transform of a function concentrated on a smooth closed surface. Part 2 The WKB method for ordinary differential equations: the asymptotic behaviour of solutions of a homogeneous equation; the scattering problem; the asymptotic behaviour of solutions of boundary-value problems. Part 3 Partial differential equations of the first order and characteristics for equations of higher order: quasilinear partial differential equations of the first order; general partial differential equations of the first order; the Hamilton-Jacobi equation; example propagation of light waves in an inhomogenous medium; characteristic surfaces for differential operators of high order, connection with the well-posedness of the Cauchy problem; the search for characteristic surfaces. Part 4 Propagation of discontinuities, problems with rapidly oscillating data: the Leibniz formula; problems with rapidly oscillating initial data; discontinuous solutions of equations. Part 5 The Maslov canonical operator: the problem of scattering of a plane wave in an inhomogenous medium; the Lagrange maniford; the precanonical operator; the canonical operator, construction of a formal asymptotic solution; a field in an isotropic medium with parabolic wave front; more general problems. Part 6 Elliptic problems in a bounded domain: Sobolev-Slobodetskii spaces; elliptic problems; elliptic problems with a parameter; inversion of a finitely-meromorphic Fredholm family of operators. Part 7 Equations and systems with constant coefficients in Rn: equations with a non-zero characteristic polynomial; equations and systems of the type of the Helmholtz equation radiation conditions; the principle of limiting absorption. Part 8 Elliptic equations with variable coefficients and boundary-value problems in the exterior of a bounded domain: solubility and a priori estimated of solutions of exterior boundary-value problems; the principle of limiting absorption for exterior problems. Part 9 Analytic properties of the resolvent of operators that depend polynomially on a parameter: equations with constant coefficients; equations with variable coefficients and problems in the interior of a bounded domain; the asymptotic behaviour of solutions of exterior problems for small frequencies. Part 10 Short-wave asymptotic behaviour of solutions of stationary problems and the asymptotic behaviour of solutions of hyperbolic equations as t ; introduction; short-wave asymptotic behaviour as t of solutions of mixed problems. Part 11 Quasiclasical approximations in stationary scattering problems: the asymptotic behaviour of the solution of the scattering problem and the amplitude of the scattering; proof of theorems 1 and 2. Part contents...


    Vainberg, B